Integral operators on new families of meromorphic functions of complex order

Abstract

In this article, we define and investigate new families of certain subclasses of meromorphic functions of complex order. Considering the new subclasses, several properties for certain integral operators are derived.

2010 Mathematics Subject Classification: 30C45.

1 Introduction

Let Σ denote the class of functions of the form:

$$f\left(z\right)=\frac{1}{z}+\sum _{n=0}^{\infty }{a}_n{z}^n,$$

(1.1)

which are analytic in the punctured open unit disk

$${\mathbb{U}}^{*}=\left\{z\in ℂ:0<\left|z\right|<1\right\}=\mathbb{U}\backslash \left\{0\right\},$$

(1.2)

where $\mathbb{U}$ is the open unit disk $\mathbb{U}=\left\{z\in ℂ:\left|z\right|<1\right\}.$.

We say that a function f ∈ Σ is meromorphic starlike of order δ (0 ≤ δ < 1), and belongs to the class Σ*(δ), if it satisfies the inequality:

$$-\Re \left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\delta .$$

(1.3)

A function f ∈ Σ is a meromorphic convex function of order δ (0 ≤ δ < 1), if f satisfies the following inequality:

$$-\Re \left(1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\right)>\delta ,$$

(1.4)

and we denote this class by Σ _k (δ).

For f ∈ Σ, Wang et al. [1] and Nehari and Netanyahu [2] introduced and studied the subclass Σ _N (β) of Σ consisting of functions f(z) satisfying

$$-\Re \left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+1\right)<\beta \left(\beta >1,z\in \mathbb{U}\right).$$

(1.5)

Let $\mathcal{A}$ denote the class of functions f of the form $f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n$, which are analytic in the open unit disk $\mathbb{U}$.

Analogous to several subclasses [3–10] of analytic functions of $\mathcal{A}$, we define the following subclasses of Σ.

Definition 1.1 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class ${\mathrm{\Sigma }}_b^{\star }\left(\delta \right)$ if, and only if, f satisfies

$$\Re \left\{1-\frac{1}{b}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+1\right)\right\}>\delta ,$$

(1.6)

where $b\in ℂ\backslash \left\{0\right\}$, 0 ≤ δ < 1.

Definition 1.2 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class ΣK _b (δ) if, and only if, f satisfies

$$\Re \left\{1-\frac{1}{b}\left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+2\right)\right\}>\delta ,$$

(1.7)

where b ∈ C\{0}, 0 ≤ δ < 1. We note that f ∈ Σ K _b (δ) if, and only if, $-z{f}^{\prime }\in {\mathrm{\Sigma }}_b^{\star }\left(\delta \right)$.

Furthermore, the classes

$${\mathrm{\Sigma }}_1^{\star }\left(\delta \right)\equiv {\mathrm{\Sigma }}^{\star }\left(\delta \right),\mathrm{\Sigma }{K}_1\left(\delta \right)\equiv {\mathrm{\Sigma }}_k\left(\delta \right)$$

are the classes of meromorphic starlike functions of order δ and meromorphic convex functions of order δ in ${\mathbb{U}}^{*}$, respectively. Moreover, the classes

$${\mathrm{\Sigma }}_1^{\star }\left(0\right)\equiv {\mathrm{\Sigma }}^{\star }\left(0\right),\mathrm{\Sigma }{K}_1\left(0\right)\equiv {\mathrm{\Sigma }}_k\left(0\right)$$

are familiar classes of starlike and convex functions in ${\mathbb{U}}^{*}$, respectively.

Definition 1.3 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class ${\mathrm{\Sigma }}^{\star }\mathcal{U}\left(\alpha ,\delta ,b\right)$ if, and only if, f satisfies

$$\Re \left\{1-\frac{1}{b}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+1\right)\right\}>\alpha \left|\frac{1}{b}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+1\right)\right|+\delta ,$$

(1.8)

where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, $b\in ℂ\backslash \left\{0\right\}$.

Definition 1.4 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class $\mathrm{\Sigma }K\mathcal{U}\left(\alpha ,\delta ,b\right)$ if, and only if, f satisfies

$$\Re \left\{1-\frac{1}{b}\left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+2\right)\right\}>\alpha \left|\frac{1}{b}\left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+2\right)\right|+\delta ,$$

(1.9)

where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, $b\in ℂ\backslash \left\{0\right\}$.

We note that $f\in \mathrm{\Sigma }K\mathcal{U}\left(\alpha ,\delta ,b\right)$ if, and only if, $-z{f}^{\prime }\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\left(\alpha ,\delta ,b\right)$.

For α = 0, we have

$${\mathrm{\Sigma }}^{\star }\mathcal{U}\left(0,\delta ,b\right)\equiv {\mathrm{\Sigma }}_b^{\star }\left(\delta \right),\mathrm{\Sigma }K\mathcal{U}\left(0,\delta ,b\right)\equiv \mathrm{\Sigma }{K}_b\left(\delta \right).$$

Definition 1.5 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class ${\mathrm{\Sigma }}^{\star }\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$ if, and only if, f satisfies

$$\begin{array}{c}\left|1-\frac{1}{b}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+1\right)-2\alpha \left(\sqrt{2}-1\right)\right|<\Re \left\{\sqrt{2}\left(1-\frac{1}{b}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+1\right)\right)\right\}+2\alpha \left(\sqrt{2}-1\right),\hfill \end{array}$$

(1.10)

where α > 0, $b\in ℂ\backslash \left\{0\right\}$.

Definition 1.6 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class $\mathrm{\Sigma }K\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$ if, and only if, f satisfies

$$\begin{array}{c}\left|1-\frac{1}{b}\left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+2\right)-2\alpha \left(\sqrt{2}-1\right)\right|<\Re \left\{\sqrt{2}\left(1-\frac{1}{b}\left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+2\right)\right)\right\}+2\alpha \left(\sqrt{2}-1\right).\hfill \end{array}$$

(1.11)

where α > 0, $b\in ℂ\backslash \left\{0\right\}$.

We note that $f\in \mathrm{\Sigma }K\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$ if, and only if, $-z{f}^{\prime }\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$.

Let us also introduce the following families of new subclasses $\mathrm{\Sigma }{\mathcal{F}}_1\left(\delta ,b\right)$, $\mathrm{\Sigma }{\mathcal{F}}_2\left(\alpha ,\delta ,b\right)$, and $\mathrm{\Sigma }{\mathcal{F}}_3\left(\alpha ,b\right)$ as follows.

Definition 1.7 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class $\mathrm{\Sigma }{\mathcal{F}}_1\left(\delta ,b\right)$, if, and only if, f satisfies

$$\Re \left\{1-\frac{1}{b}\left(\frac{z\left(z{f}^{″}\left(z\right)+3f^{\prime }\left(z\right)\right)}{z{f}^{\prime }\left(z\right)+2f\left(z\right)}+1\right)\right\}>\delta .$$

(1.12)

where $b\in ℂ\backslash \left\{0\right\}$, 0 ≤ δ < 1.

We note that $f\in \mathrm{\Sigma }{\mathcal{F}}_1\left(\delta ,b\right)$ if, and only if, $z{f}^{\prime }\left(z\right)+2f\left(z\right)\in {\mathrm{\Sigma }}_b^{\star }\left(\delta \right)$.

Definition 1.8 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class $\mathrm{\Sigma }{\mathcal{F}}_2\left(\alpha ,\delta ,b\right)$ if, and only if, f satisfies

$$\Re \left\{1-\frac{1}{b}\left(\frac{z\left(z{f}^{″}\left(z\right)+3f^{\prime }\left(z\right)\right)}{z{f}^{\prime }\left(z\right)+2f\left(z\right)}+1\right)\right\}>\alpha \left|\frac{1}{b}\left(\frac{z\left(z{f}^{″}\left(z\right)+3f^{\prime }\left(z\right)\right)}{z{f}^{\prime }\left(z\right)+2f\left(z\right)}+1\right)\right|+\delta ,$$

(1.13)

where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, $b\in ℂ\backslash \left\{0\right\}$.

We note that $f\in \mathrm{\Sigma }{\mathcal{F}}_2\left(\alpha ,\delta ,b\right)$ if, and only if, $z{f}^{\prime }\left(z\right)+2f\left(z\right)\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\left(\alpha ,\delta ,b\right)$.

Definition 1.9 Let a function f ∈ Σ be analytic in ${\mathbb{U}}^{*}$. Then f is in the class $\mathrm{\Sigma }{\mathcal{F}}_3\left(\alpha ,b\right)$ if, and only if, f satisfies

$$\begin{array}{cc}\left|1-\frac{1}{b}\left(\frac{z\left(z{f}^{″}\left(z\right)+3f^{\prime }\left(z\right)\right)}{z{f}^{\prime }\left(z\right)+2f\left(z\right)}+1\right)-2\alpha \left(\sqrt{2}-1\right)\right|\hfill & <\Re \left\{\sqrt{2}\left(1-\frac{1}{b}\left(\frac{z\left(z{f}^{″}\left(z\right)+3f^{\prime }\left(z\right)\right)}{z{f}^{\prime }\left(z\right)+2f\left(z\right)}+1\right)\right)\right\}\hfill \\ +2\alpha \left(\sqrt{2}-1\right),\hfill \end{array}$$

(1.14)

where α > 0, $b\in ℂ\backslash \left\{0\right\}$.

We note that $f\in \mathrm{\Sigma }{\mathcal{F}}_3\left(\alpha ,b\right)$ if, and only if, $z{f}^{\prime }\left(z\right)+2f\left(z\right)\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$.

Recently, many authors introduced and studied various integral operators of analytic and univalent functions in the open unit disk $\mathbb{U}$ [11–21].

Most recently, Mohammed and Darus [22, 23] introduced the following two general integral operators of meromorphic functions Σ:

$${\mathcal{H}}_n\left(z\right)=\frac{1}{z^2}\underset{0}{\overset{z}{\int }}{\left(u{f}_1\left(u\right)\right)}^{{\gamma }_1}\dots {\left(u{f}_n\left(u\right)\right)}^{{\gamma }_n}du,$$

(1.15)

and

$${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)=\frac{1}{z^2}\underset{0}{\overset{z}{\int }}{\left(-u^2{f}_1^{\prime }\left(u\right)\right)}^{{\gamma }_1}\dots {\left(-u^2{f}_n^{\prime }\left(u\right)\right)}^{{\gamma }_n}du.$$

(1.16)

Goyal and Prajapat [24] obtained the following results for f ∈ Σ to be in the class Σ*(δ), 0 ≤ δ < 1.

Corollary 1.1 If f ∈ Σ satisfies the following inequality

$$\left|\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}-\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right|<\frac{\left(1-\delta \right)\left(3-\delta \right)}{2-\delta },0\le \delta <1,$$

(1.17)

then f ∈ Σ*(δ).

Corollary 1.2 If f ∈ Σ satisfies the following inequality

$$\left|\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}-\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+1\right|<\frac{1}{2},$$

(1.18)

then f ∈ Σ*.

Corollary 1.3 If f ∈ Σ satisfies the following inequality

$$\Re \left\{\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\left(\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)}-\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}-1\right)\right\}>-\frac{1}{2},$$

(1.19)

then f ∈ Σ*.

In this article, we derive several properties for the integral operators ${\mathcal{H}}_n\left(z\right)$ and ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)$ of the subclasses given by (1.5) and Definitions 1.1 to 1.6.

2 Some properties for ${\mathcal{H}}_n\left(z\right)$

In this section, we investigate some properties for the integral operator ${\mathcal{H}}_n\left(z\right)$ defined by (1.15) of the subclasses given by (1.5), Definitions 1.1, 1.3, and 1.5.

Theorem 2.1 For i ∈ {1,..., n}, let γ _i > 0, f _i ∈ Σ and

$$\left|\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}-2\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}\right|<\frac{\left(1-\delta \right)\left(3-\delta \right)}{2-\delta },\left(0\le \delta <1\right).$$

(2.1)

If

$$\sum _{i=1}^n{\gamma }_i>\frac{2}{1-\delta },$$

(2.2)

then ${\mathcal{H}}_n\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, where β > 1.

Proof On successive differentiation of ${\mathcal{H}}_n\left(z\right)$, which is defined in (1.5), we get

$$z^2{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)+2z{\mathcal{H}}_n\left(z\right)={\left(z{f}_1\left(z\right)\right)}^{{\gamma }_1}\dots {\left(z{f}_n\left(z\right)\right)}^{{\gamma }_n},$$

(2.3)

and

$$\begin{array}{c}{z}^2{\mathcal{H}}_n^{{″}^{}}\left(z\right)+4z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)+2{\mathcal{H}}_n\left(z\right)=\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)+f_i\left(z\right)}{z{f}_i\left(z\right)}\right)\left[{\left(z{f}_1\left(z\right)\right)}^{{\gamma }_1}\dots {\left(z{f}_n\left(z\right)\right)}^{{\gamma }_n}\right].\hfill \end{array}$$

(2.4)

Then from (2.3) and (2.4), we obtain

$$\frac{z^2{\mathcal{H}}_n^{{″}^{}}\left(z\right)+4z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)+2{\mathcal{H}}_n\left(z\right)}{z^2{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)+2z{\mathcal{H}}_n\left(z\right)}=\sum _{i=1}^n{\gamma }_i\left(\frac{f_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+\frac{1}{z}\right).$$

(2.5)

By multiplying (2.5) with z yield

$$\frac{z^2{\mathcal{H}}_n^{″}\left(z\right)+4z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}=\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right).$$

(2.6)

This is equivalent to

$$\frac{z^2{\mathcal{H}}_n^{{″}^{}}\left(z\right)+3z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1=\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right).$$

(2.7)

Therefore, we have

$$\frac{-\left(\frac{z{\mathcal{H}}_n^{{″}^{}}\left(z\right)}{{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}+1\right)-2}{1+\frac{2{\mathcal{H}}_n\left(z\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}}=-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)+1.$$

(2.8)

Then, we easily get

$$\begin{array}{cc}-\left(\frac{z{\mathcal{H}}_n^{{″}^{}}\left(z\right)}{{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}+1\right)\hfill & =\left(\frac{2{\mathcal{H}}_n\left(z\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)+1\right]\hfill \\ +\sum _{i=1}^n{\gamma }_i\left(-\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}\right)+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(2.9)

Taking real parts of both sides of (2.9), we obtain

$$\begin{array}{cc}-\Re \left(\frac{z{\mathcal{H}}_n^{{″}^{}}\left(z\right)}{{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}+1\right)\hfill & =\Re \left\{\left(\frac{2{\mathcal{H}}_n\left(z\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)+1\right]\right\}\hfill \\ +\sum _{i=1}^n{\gamma }_i\Re \left(-\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}\right)+3-\sum _{i=1}^n{\gamma }_i\hfill \\ \le \left|\left(\frac{2{\mathcal{H}}_n\left(z\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)+1\right]\right|\hfill \\ +\sum _{i=1}^n{\gamma }_i\Re \left(-\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}\right)+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(2.10)

Let

$$\begin{array}{cc}\beta \hfill & =\left|\left(\frac{2{\mathcal{H}}_n\left(z\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)+1\right]\right|\hfill \\ +\sum _{i=1}^n{\gamma }_i\Re \left(-\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}\right)+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(2.11)

Since $\left|\left(\frac{2{\mathcal{H}}_n\left(z\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)+1\right]\right|>0$, applying Corollary 1.1, we have

$$\beta >\delta \sum _{i=1}^n{\gamma }_i+3-\sum _{i=1}^n{\gamma }_i=3-\left(1-\delta \right)\sum _{i=1}^n{\gamma }_i.$$

(2.12)

Then, by the hypothesis (2.2), we have β > 1. Therefore, ${\mathcal{H}}_n\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, where β > 1. This completes the proof.   □

Letting δ = 0 in Theorem 2.1, we have

Corollary 2.2 For i ∈ {1,..., n}, let γ _i > 0, f _i ∈ Σ and

$$\left|\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}-2\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}\right|<\frac{3}{2}.$$

(2.13)

If

$$\sum _{i=1}^n{\gamma }_i>2,$$

(2.14)

then ${\mathcal{H}}_n\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, where β > 1.

Making use of (2.11), Corollary 1.2 and Corollary 1.3, one can prove the following results.

Theorem 2.3 For i ∈ {1,..., n}, let γ _i > 0, f _i ∈ Σ and

$$\left|\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}-\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right|<\frac{1}{2}.$$

(2.15)

If

$$\sum _{i=1}^n{\gamma }_i>2,$$

(2.16)

then ${\mathcal{H}}_n\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, where β > 1.

Theorem 2.4 For i ∈ {1,..., n}, let γ _i > 0, f _i ∈ Σ and

$$\Re \left\{\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}\left(\frac{2z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}-\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}-1\right)\right\}>-\frac{1}{2}.$$

(2.17)

If

$$\sum _{i=1}^n{\gamma }_i>2,$$

(2.18)

then ${\mathcal{H}}_n\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, where β > 1.

Theorem 2.5 For i ∈ {1,..., n}, let γ _i > 0 and $f_i\in {\mathrm{\Sigma }}_b^{\star }\left({\delta }_i\right)$ (0 ≤ δ < 1 and $b\in ℂ\backslash \left\{0\right\}$).

If

$$0<\sum _{i=1}^n{\gamma }_i\left(1-{\delta }_i\right)\le 1,$$

(2.19)

then ${\mathcal{H}}_n\left(z\right)$ is in the class $\mathrm{\Sigma }{\mathcal{F}}_1\left(\mu ,b\right),\mu =1-\sum _{i=1}^n{\gamma }_i\left(1-{\delta }_i\right)$.

Proof From (2.7), we have

$$\frac{z\left(z{\mathcal{H}}_n^{{″}^{}}\left(z\right)+3{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)\right)}{z{\mathcal{H}}_n^{{\prime }^{}}\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1=\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right).$$

(2.20)

Equivalently, (2.20) can be written as

$$1-\frac{1}{b}\left\{\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right\}=\sum _{i=1}^n{\gamma }_i\left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}+1-\sum _{i=1}^n{\gamma }_i.$$

(2.21)

Taking the real part of both terms of the last expression, we have

$$\Re \left\{1-\frac{1}{b}\left(\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right)\right\}=\sum _{i=1}^n{\gamma }_i\Re \left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}+1-\sum _{i=1}^n{\gamma }_i.$$

(2.22)

Since $f_i\in {\mathrm{\Sigma }}_b^{\star }\left({\delta }_i\right)$, hence

$$\Re \left\{1-\frac{1}{b}\left(\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right)\right\}>\sum _{i=1}^n{\gamma }_i{\delta }_i+1-\sum _{i=1}^n{\gamma }_i.$$

(2.23)

so that

$$\Re \left\{1-\frac{1}{b}\left(\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right)\right\}>1-\sum _{i=1}^n{\gamma }_i\left(1-{\delta }_i\right).$$

(2.24)

Then ${\mathcal{H}}_n\left(z\right)\in \mathrm{\Sigma }{\mathcal{F}}_1\left(\mu ,b\right),\mu =1-\sum _{i=1}^n{\gamma }_i\left(1-{\delta }_i\right)$.

Now, adopting the techniques used by Breaz et al. [11] and Bulut [21], we prove the following two theorems.

Theorem 2.6 For i ∈ {1,..., n}, let γ _i > 0 and $f_i\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\left(\alpha ,\delta ,b\right)$ (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and $b\in ℂ\backslash \left\{0\right\}$). If

$$\sum _{i=1}^n{\gamma }_i\le 1,$$

(2.25)

then ${\mathcal{H}}_n\left(z\right)$ is in the class $\mathrm{\Sigma }{\mathcal{F}}_2\left(\alpha ,\delta ,b\right)$.

Proof Since $f_i\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\left(\alpha ,\delta ,b\right)$, it follows from Definition 1.3 that

$$\Re \left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}>\alpha \left|\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right|+\delta .$$

(2.26)

Considering Definition 1.8 and with the help of (2.21), we obtain

$$\begin{array}{c}\Re \left\{1-\frac{1}{b}\left(\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right)\right\}-\alpha \left|\frac{1}{b}\left(\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right)\right|-\delta \hfill \\ =1-\sum _{i=1}^n{\gamma }_i+\sum _{i=1}^n{\gamma }_i\Re \left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}-\alpha \left|\sum _{i=1}^n{\gamma }_i\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right|-\delta \hfill \\ \ge 1-\sum _{i=1}^n{\gamma }_i+\sum _{i=1}^n{\gamma }_i\Re \left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}-\alpha \sum _{i=1}^n{\gamma }_i\left|\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right|-\delta \hfill \\ >1-\sum _{i=1}^n{\gamma }_i+\sum _{i=1}^n{\gamma }_i\left\{\alpha \left|\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_{{\prime }^{}}\left(z\right)}+1\right)\right|+\delta \right\}-\alpha \sum _{i=1}^n{\gamma }_i\left|\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right|-\delta \hfill \\ =\left(1-\delta \right)\left(1-\sum _{i=1}^n{\gamma }_i\right)\ge 0.\hfill \end{array}$$

(2.27)

This completes the proof.   □

Theorem 2.7 For i ∈ {1,..., n}, let γ _i > 0 and $f_i\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$ (α > 0 and $b\in ℂ\backslash \left\{0\right\}$). If

$$\sum _{i=1}^n{\gamma }_i\le 1,$$

(2.28)

then ${\mathcal{H}}_n\left(z\right)$ is in the class $\mathrm{\Sigma }{\mathcal{F}}_3\left(\alpha ,b\right)$.

Proof Since $f_i\in {\mathrm{\Sigma }}^{\star }\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$, it follows from Definition 1.5 that

$$\Re \left\{\sqrt{2}\left(1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right)\right\}+2\alpha \left(\sqrt{2}-1\right)-\left|1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)-2\alpha \left(\sqrt{2}-1\right)\right|>0.$$

(2.29)

Considering this inequality and (2.21), we obtain

$$\begin{array}{c}\Re \left\{\sqrt{2}\left(1-\frac{1}{b}\left(\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right)\right)\right\}+2\alpha \left(\sqrt{2}-1\right)-\left|1-\frac{1}{b}\left(\frac{z\left(z{\mathcal{H}}_n^{″}\left(z\right)+3{\mathcal{H}}_n^{\prime }\left(z\right)\right)}{z{\mathcal{H}}_n^{\prime }\left(z\right)+2{\mathcal{H}}_n\left(z\right)}+1\right)\right.\hfill \\ \left.-2\alpha \left(\sqrt{2}-1\right)\right|\hfill \\ =\Re \left\{\sqrt{2}\left[1-\sum _{i=1}^n{\gamma }_i\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right]\right\}+2\alpha \left(\sqrt{2}-1\right)-\left|1-\sum _{i=1}^n{\gamma }_i\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right.\hfill \\ \left.-2\alpha \left(\sqrt{2}-1\right)\right|\hfill \\ =\sqrt{2}-\sqrt{2}\sum _{i=1}^n{\gamma }_i\Re \frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)+2\alpha \left(\sqrt{2}-1\right)-\left|1-\sum _{i=1}^n{\gamma }_i\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right.\hfill \\ \left.-2\alpha \left(\sqrt{2}-1\right)\right|\hfill \\ =\sqrt{2}+\sqrt{2}\sum _{i=1}^n{\gamma }_i\Re \left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}-\sqrt{2}\sum _{i=1}^n{\gamma }_i+2\alpha \left(\sqrt{2}-1\right)\hfill \\ -\left|1+\sum _{i=1}^n{\gamma }_i\left[1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)-2\alpha \left(\sqrt{2}-1\right)\right]-\sum _{i=1}^n{\gamma }_i+2\alpha \left(\sqrt{2}-1\right)\sum _{i=1}^n{\gamma }_i\right.\hfill \\ -\left.2\alpha \left(\sqrt{2}-1\right)\right|\hfill \\ =\sqrt{2}\left(1-\sum _{i=1}^n{\gamma }_i\right)+2\alpha \left(\sqrt{2}-1\right)+\sqrt{2}\sum _{i=1}^n{\gamma }_i\Re \left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}\hfill \\ -\left|\left[1-2\alpha \left(\sqrt{2}-1\right)\right]\left(1-\sum _{i=1}^n{\gamma }_i\right)+\sum _{i=1}^n{\gamma }_i\left[1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)-2\alpha \left(\sqrt{2}-1\right)\right]\right|\hfill \\ \ge \sqrt{2}\left(1-\sum _{i=1}^n{\gamma }_i\right)+2\alpha \left(\sqrt{2}-1\right)+\sqrt{2}\sum _{i=1}^n{\gamma }_i\Re \left\{1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right\}\hfill \\ -\sum _{i=1}^n{\gamma }_i\left|1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)-2\alpha \left(\sqrt{2}-1\right)\right|-\left|1-2\alpha \left(\sqrt{2}-1\right)\right|\left(1-\sum _{i=1}^n{\gamma }_i\right)\hfill \\ =\sum _{i=1}^n{\gamma }_i\left\{\Re \sqrt{2}\left[1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)\right]+2\alpha \left(\sqrt{2}-1\right)-\left|1-\frac{1}{b}\left(\frac{z{f}_i^{{\prime }^{}}\left(z\right)}{f_i\left(z\right)}+1\right)-2\alpha \left(\sqrt{2}-1\right)\right|\right\}\hfill \\ +\sqrt{2}\left(1-\sum _{i=1}^n{\gamma }_i\right)+2\alpha \left(\sqrt{2}-1\right)-2\alpha \left(\sqrt{2}-1\right)\sum _{i=1}^n{\gamma }_i-\left|1-2\alpha \left(\sqrt{2}-1\right)\right|\left(1-\sum _{i=1}^n{\gamma }_i\right)\hfill \\ >\left[\sqrt{2}+2\alpha \left(\sqrt{2}-1\right)-\left|1-2\alpha \left(\sqrt{2}-1\right)\right|\right]\left(1-\sum _{i=1}^n{\gamma }_i\right)\hfill \\ >\left(1-\sum _{i=1}^n{\gamma }_i\right)min\left\{\left(\sqrt{2}-1\right)\left(1+4\alpha \right),\sqrt{2}+1\right\}\ge 0.\hfill \end{array}$$

This completes the proof.   □

3 Some properties for ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)$

In this section, we investigate some properties for the integral operator ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)$ defined by (1.16) of subclasses given by (1.5), Definitions 1.2, 1.4, and 1.6.

Theorem 3.1 For i ∈ {1,..., n}, let γ _i > 0, f _i ∈ Σ and

$$\sum _{i=1}^n{\gamma }_i>\frac{2}{1-\delta },\left(0\le \delta <1\right).$$

(3.1)

If $f_i\in {\mathrm{\Sigma }}_k\left(\delta \right)$, then ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, β > 1.

Proof On successive differentiation of ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)$, which is defined in (1.16), we have

$$2z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)+z^2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)={\left(-z^2{f}_1^{{\prime }^{}}\left(z\right)\right)}^{{\gamma }_1}\dots {\left(-z^2{f}_n^{{\prime }^{}}\left(z\right)\right)}^{{\gamma }_n},$$

(3.2)

and

$$\begin{array}{c}{z}^2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{″}\left(z\right)+4z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)+2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)\hfill \\ =\sum _{i=1}^n{\gamma }_i\left(\frac{f_i^{″}\left(z\right)}{f_i^{\prime }\left(z\right)}+\frac{2}{z}\right)\left[{\left(-z^2{f}_1^{\prime }\left(z\right)\right)}^{{\gamma }_1}\dots {\left(-z^2{f}_n^{\prime }\left(z\right)\right)}^{{\gamma }_n}\right].\hfill \end{array}$$

(3.3)

Then from (3.2) and (3.3), we obtain

$$\frac{z^2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{″}\left(z\right)+4z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)+2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z^2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)+2z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}=\sum _{i=1}^n{\gamma }_i\left(\frac{{f_i}^{″}\left(z\right)}{{f_i}^{\prime }\left(z\right)}+\frac{2}{z}\right).$$

(3.4)

By multiplying (3.4) with z yields,

$$\frac{z^2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{″}\left(z\right)+4z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)+2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)+2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}=\sum _{i=1}^n{\gamma }_i\left(\frac{z{f_i}^{″}\left(z\right)}{{f_i}^{\prime }\left(z\right)}+2\right).$$

(3.5)

that is equivalent to

$$\frac{z^2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{″}\left(z\right)+3z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{\prime }\left(z\right)+2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}+1=\sum _{i=1}^n{\gamma }_i\left(\frac{z{f_i}^{″}\left(z\right)}{{f_i}^{\prime }\left(z\right)}+2\right).$$

(3.6)

Therefore, we have

$$\frac{-\left(\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{″}^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}+1\right)-2}{1+\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}}=-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+2\right)+1.$$

(3.7)

so that

$$\begin{array}{cc}-\left(\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{″}^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}+1\right)\hfill & =\left(\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+2\right)+1\right]\hfill \\ +\sum _{i=1}^n{\gamma }_i\left\{-\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right\}+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(3.8)

Taking the real parts of both terms of the last expression, we obtain

$$\begin{array}{cc}-\Re \left(\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{″}^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}+1\right)\hfill & =\Re \left\{\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+2\right)+1\right]\right\}\hfill \\ +\sum _{i=1}^n{\gamma }_i\Re \left\{-\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right\}+3-\sum _{i=1}^n{\gamma }_i\hfill \\ \le \left|\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+2\right)+1\right]\right|\hfill \\ +\sum _{i=1}^n{\gamma }_i\Re \left\{-\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right\}+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(3.9)

Let

$$\beta =\left|\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+2\right)+1\right]\right|+\sum _{i=1}^n{\gamma }_i\Re \left\{-\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right\}+3-\sum _{i=1}^n{\gamma }_i.$$

(3.10)

Since $\left|\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+2\right)+1\right]\right|>0$, f _i ∈ Σ _k (δ), we get

$$\beta >3-\left(1-\delta \right)\sum _{i=1}^n{\gamma }_i,$$

(3.11)

which, in light of the hypothesis (3.1), yields β > 1.

Therefore, ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, β > 1. This completes the proof.   □

Theorem 3.2 For i ∈ {1,..., n}, let γ _i > 0, f _i ∈ Σ and

$$1<\sum _{i=1}^n{\gamma }_i<2.$$

(3.12)

If ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)\in {\mathrm{\Sigma }}^{\star }\left(\delta \right)$, then ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, β > 1.

Proof It follows from (3.7) that

$$\begin{array}{cc}-\left(\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{″}^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}+1\right)\hfill & =\left(1+\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right]\hfill \\ +2\left(\sum _{i=1}^n{\gamma }_i-1\right)\left(-\frac{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(3.13)

Taking the real parts of both terms of the last expression, we obtain

$$\begin{array}{cc}-\Re \left(\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{″}^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}+1\right)\hfill & =\Re \left\{\left(1+\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right]\right\}\hfill \\ +2\left(\sum _{i=1}^n{\gamma }_i-1\right)\Re \left(-\frac{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(3.14)

Thus, we have

$$\begin{array}{cc}-\Re \left(\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{″}^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}+1\right)\hfill & =\Re \left\{\left(1+\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right]\right\}\hfill \\ +2\left(\sum _{i=1}^n{\gamma }_i-1\right)\Re \left(-\frac{1}{\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}}\right)+3-\sum _{i=1}^n{\gamma }_i\hfill \\ \le \left|\left(1+\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right]\right|\hfill \\ +2\left(\sum _{i=1}^n{\gamma }_i-1\right)\frac{\Re \left(-\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}\right)}{{\left|\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}{H_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}\right|}^2}+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(3.15)

Let

$$\begin{array}{cc}\beta \hfill & =\left|\left(1+\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right]\right|\hfill \\ +2\left(\sum _{i=1}^n{\gamma }_i-1\right)\frac{\Re \left(-\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}\right)}{{\left|\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}\right|}^2}+3-\sum _{i=1}^n{\gamma }_i.\hfill \end{array}$$

(3.16)

Since $\left|\left(1+\frac{2{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}\right)\left[-\sum _{i=1}^n{\gamma }_i\left(\frac{z{f}_i^{{″}^{}}\left(z\right)}{f_i^{{\prime }^{}}\left(z\right)}+1\right)\right]\right|>0$ and ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)\in {\mathrm{\Sigma }}^{\star }\left(\delta \right)$, we have

$$\beta >2\left(\sum _{i=1}^n{\gamma }_i-1\right)\frac{\delta }{{\left|\frac{z{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}^{{\prime }^{}}\left(z\right)}{{\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)}\right|}^2}+3-\sum _{i=1}^n{\gamma }_i.>3-\sum _{i=1}^n{\gamma }_i.$$

(3.17)

Then, by the hypothesis (3.12), we see that β > 1. Therefore, ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)\in {\mathrm{\Sigma }}_N\left(\beta \right)$, β > 1. This completes the proof.   □

Now, using the method given in the proofs of Theorems 2.5, 2.6, and 2.7, one can prove the following results:

Theorem 3.3 For i ∈ {1,..., n}, let γ _i > 0 f _i ∈ ΣK _b (δ _i ) (0 ≤ δ < 1 and $b\in ℂ\backslash \left\{0\right\}$)). If

$$0<\sum _{i=1}^n{\gamma }_i\left(1-{\delta }_i\right)\le 1,$$

(3.18)

then ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)$ is in the class $\mathrm{\Sigma }{\mathcal{F}}_1\left(\mu ,b\right),\mu =1-\sum _{i=1}^n{\gamma }_i\left(1-{\delta }_i\right)$.

Theorem 3.4 For i ∈ {1,..., n}, let γ _i > 0 and $f_i\in \mathrm{\Sigma }K\mathcal{U}\left(\alpha ,\delta ,b\right)$ (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and $b\in ℂ\backslash \left\{0\right\}$)). If

$$\sum _{i=1}^n{\gamma }_i\le 1,$$

(3.19)

then ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)$ is in the class $\mathrm{\Sigma }{\mathcal{F}}_2\left(\alpha ,\delta ,b\right)$.

Theorem 3.5 For i ∈ {1,..., n}, let γ _i > 0 and $f_i\in \mathrm{\Sigma }K\mathcal{U}\mathcal{H}\left(\alpha ,b\right)$ (α ≥ 0, and $b\in ℂ\backslash \left\{0\right\}$)). If

$$\sum _{i=1}^n{\gamma }_i\le 1,$$

(3.20)

then ${\mathcal{H}}_{{\gamma }_1,\dots ,{\gamma }_n}\left(z\right)$ is in the class $\mathrm{\Sigma }{\mathcal{F}}_3\left(\alpha ,\delta ,b\right)$.