Abstract
The main object of this paper is to investigate some subordination results of certain subclasses of multivalent analytic functions which are defined by a generalized fractional differintegral operator. Inclusion relations for functions in the class \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \) and the images of these functions by the generalized Bernardi-Libera-Livingston integral operator are also considered.
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Introduction
Denote the class consisting of analytic and multivalent functions in the open unit disc \(\mathbb {U}=\{z\in \mathbb {C}:\left \vert z\right \vert <1\}\) of the form:
by \(\mathcal {A}(p).\) We note that \(\mathcal {A}(1)=\mathcal {A}.\)
Consider the first-order differential subordination
where the symbol ≺ stands for subordination of two analytic functions in \(\mathbb {U\ }\)(see [1, 2]). A univalent function q is called dominant, if φ(z)≺q(z) for all analytic functions φ that satisfy this differential subordination. A dominant \(\widetilde {q}\) is called the best dominant, if \(\widetilde {q}(z)\prec q(z)\) for all dominant q. For \(f\in \mathcal {A}(p)\), the qth order derivative of f(z) could be written as
where
Let
be the well-known generalized hypergeometric function for complex parameters \(a_{1},...,a_{q}\,,\ b_{1},...,b_{s}\ \ (b_{j}\notin \mathbb {Z}_{0}^{-}=\{0,-1,-2,...\};\ j=1,2,...,s)\ \)and (λ)ν is the Pochhammer symbol defined by
In addition, if we put p=2, q=1, a1=a, a2=b, b1=c in (3), we get the (Gaussian) hypergeometric function 2F1(a,b;c;z)(c≠0,−1,−2,…) which satisfies (see [3])
and
We will recall some definitions which will be used in our paper.
Definition 1
[4–12]. Assume that 0≤λ<1 and \(\mu,\eta \in \mathbb {R}\). Then, in terms of 2F1, the generalized fractional derivative operator for \(f\in \mathcal {A}(p)\) is defined by
where f is an analytic function in a simply-connected region of the complex z-plane containing the origin with the order f(z)=O(|z|ε), z→0 when ε> max{0,μ−η}−1 and the multiplicity of (z−ζ)−λ is removed by requiring log(z−ζ)to be real when z−ζ>0.
Remark 1
We note that
(i) \(J_{0,z}^{\lambda,\mu,\eta,p}\left \{ z^{p+n}\right \} =\frac {\Gamma (p+n+1)\Gamma (p+n+1-\mu +\eta)}{\Gamma (p+n+1-\mu)\Gamma (p+n+1-\lambda +\eta)}z^{p+n-\mu }\ \left (n\geq 1\right),\)
(ii) \(J_{0,z}^{\lambda,\lambda,\eta,p}f(z)=D_{z}^{\lambda }f(z)\ \)(see [13]).
Goyal and Prajapat [14] (see also [4–12]) defined the operator \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}:\mathcal {A}(p)\rightarrow \mathcal {A}(p) \left (0\leq \lambda <1,\;\mu < p+1,\;\eta >\max \{\lambda,\mu \}-p-1\right),\ \)by
where the symbol ∗ stands for convolution of two power series and \(f\in \mathcal {A}(p)\). It is easy to check that
In this paper, we define the higher order derivative of \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\ \)as follows:
From (9), we have
We say that \(f\in \mathcal {A}(p)\) is in the class \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \)if
\(0\leq \lambda <1,\;\mu < p+1,\;\eta >\max \{\lambda,\mu \}-p-1,\ 0\leq \zeta < p-q,\ -1\leq B<A\leq 1,\ p\in \mathbb {N},\ q\in \mathbb {N}_{0}\ {and}\ p>q+\zeta.\ \)Denoting by \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right), \) the class of functions \(f\left (z\right) \in \mathcal {A}(p)\ \)which satisfies
Preliminaries
To prove our main results, we shall need the following definition and lemmas.
Definition 2
[2]. Denote the set of all functions f that are analytic and univalent on \(\overline {\mathbb {U}}\setminus E(f)\ \)by \(\mathcal {Q}\), where
and are such that f′(ς)≠0 for \(\varsigma \in \partial \mathbb {U}\setminus E(f)\).
Lemma 1
[15]. Let h(z) be analytic and convex (univalent) function in \(\mathbb {U}\) with h(0)=1. Also let ϕ given by
be analytic in \(\mathbb {U}\). If
then
and ψ is the best dominant of (13).
Lemma 2
[16]. Let h be a convex functions with
If p(z) is analytic in \(\mathbb {U\ }{with}\ p(0)=h(0),\ \)then
The class of star-like (and normalized) functions of order α in \(\mathbb {U},\ \)is
Also in [17], if β>0 and β+γ>0, for a given \(\alpha \in \left [ -\frac {\gamma }{\beta },1\right),\) we define the order of starlikeness of the class \(\mathrm {I}_{\beta,\gamma }\left [ \mathcal {S}^{\ast }\left (\alpha \right) \right ]\ \)by the largest number 𝜗(α;β,γ) such that\(\ \mathrm {I}_{\beta,\gamma }\left [ \mathcal {S}^{\ast }\left (\alpha \right) \right ] \subset \mathcal {S}^{\ast }\left (\vartheta \right),\ \)where Iβ,γ is given by
Lemma 3
[17]. Let β>0, β+γ>0 and consider Iβ,γ defined by (14). If \(\alpha \in \left [-\frac {\gamma }{\beta },1\right),\ \)then the order of starlikeness of the class \(\mathrm {I}_{\beta,\gamma }\left [ \mathcal {S}^{\ast }\left (\alpha \right) \right ] \ \)is given by the number \(\vartheta \left (\alpha ;\beta,\gamma \right) =\inf \left \{ \mathfrak {R}\left (q(z)\right):z\in \mathbb {U}\right \},\ \)where
Moreover, if \(\alpha \in \left [ \alpha _{0},1\right),\ {where}\ \alpha _{0}=\max \left \{ \frac {\beta -\gamma -1}{2\beta };-\frac {\gamma }{\beta }\right \}\ {and}\ g=\mathrm {I}_{\beta,\gamma }\left (f\right)\) with \(f\in \mathcal {S}^{\ast }\left (\alpha \right),\ \)then
where
Subordination and Inclusion theorems involving \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left (q\right) }\)
We assume throughout this paper unless otherwise mentioned that\(\ p\in \mathbb {N},\ 0\leq \lambda <1,\;\mu < p,\;\eta >\max \{\lambda,\mu \}-p-1,\ -1\leq B<A\leq 1,\ 0\leq \zeta < p-q,\ \xi <1,\ \sigma >0,\ 0< c\leq 1\) and the powers are considered principal ones.
Theorem 1
Assume that \(1\leq q\leq p\ {and}\ f\left (z\right)\in \mathcal {A}\left (p\right) \) satisfy
then
where
is the best dominant of (15). Furthermore,
where
The estimate in (17) is the best possible.
Proof
Putting
then ϕ(z) is analytic in \(\mathbb {U}\). After some computations, we get
where the influence of \(h(z) = \sqrt {1+cz}\) under certain values of c is illustrated by Fig. 1. To apply Lemma 1, it suffies to show that h(z) is convex, therefore for z=reiθ, r∈(0,1), θ∈[−π,π], we have
and
This implies that h is convex in \(\mathbb {U}\).
Now, by using Lemma 1 (with n=1) and making a change of variables followed by the use of (4) and (5), we deduce that
this proves (15). Next, it is enough to show that
Indeed
Setting
which is a positive measure on the closed interval [0,1], we get
so that
Letting r→1− in the above inequality, we obtain (17). To show that the result in (17) is sharp, let us suppose that
that is
From (15), we have
and so
which implies that M≤M1, that is, M cannot be decreased and the estimate in (17) is the best possible. □
For \(f\in \mathcal {A}(p)\) the generalized Bernardi-Libera-Livingston integeral operator Fp,υ is defined by (see [18]):
Lemma 4
If \(f\in \mathcal {A}\left (p\right),\ \)then (i) \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}\left (F_{p,\upsilon }f\right) =F_{p,\upsilon }\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f\right),\)
(ii)
(iii)
Proof
Since \(f(z)\in \mathcal {A}(p)\), then
and
Now, the first part of this lemma follows. Also, the recurrence relation of Fp,υ is given by
If we replace f(z) by \(\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\ \)and using the first part of this lemma, we get (20). If we differentiate (20) q-times, we obtain (21). □
Theorem 2
Suppose that \(1\leq q\leq p\ {and}\ f\left (z\right)\in \mathcal {A}(p)\) satisfy
where Fp,υ defined by (19), then
where φ(z) given by
is the best dominant of (23). Further,
where
The result is the best possible.
Proof
Taking
then Θ is analytic in \(\mathbb {U}\). After some calculations, we have
By employing the same technique that was used in proving Theorem 1, the remaining part of the theorem can be proved. □
Theorem 3
Let\(\ q\in \mathbb {N}_{0}\ {and}\ p>q+\zeta.\ \)If \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right),\ \)then \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta,\xi \right) \ \)for |z|<R(p,q,μ,ζ,ξ) where
and
Proof
Assume that \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right) \) and
then, u(z) is analytic in \(\mathbb {U}\) with \(u(0)=1,\ \mathfrak {R}\left \{ u\left (z\right) \right \} >\xi.\) After some computations, we have
Letting \(v(z)=\frac {u(z)-\xi }{1-\xi },\ \)then, v(0)=1 with \(\mathfrak {R}\left \{ v\left (z\right) \right \} >0.\ \)Substituting in (28), we obtain
and so
Applying the following well-known estimate [19]:
for n=1, we get
It is easily seen that t(r) is positive, if |z|<R(p,q,μ,ζ,ξ), where R is given by (26). □
Theorem 4
Let \(f(z)\in \mathcal {A}(p),\ p>\mu,\ \gamma >0\ \)and
then
The result is sharp.
Proof
From (8), (29) may be written as
or equivalently,
Letting
then, we can express (30) as
or equivalently,
To show that the result is sharp, let
and so
It is easy to check that K(z) satisfies (29) and
as z→1−. This ends our proof. □
Theorem 5
Consider that\(\ q\in \mathbb {N}_{0},\ p>q+\zeta \ {and}\ \)
(i) Suppose that \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast }:=\mathbb {U}\backslash \{0\},\ \)then
(ii) Also, assuming that
then
where the bound
is the best possible
Proof
Let \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta ;A,B\right) \ \)and
Since \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast },\ \) then G(z) is analytic in \(\mathbb {U}\) with G(0)=0 and G′(0)=1. Differentiating both sides of (35) logarithmically, we get
Differentiating both sides of (37) logarithmically, we get
Combining this identity together with \(f\left (z\right)\in \mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta ;A,B\right),\ \)we obtain
We will use Lemma 2 for \(\widetilde {\beta }=\left (p-q-\zeta \right),\ \widetilde {\gamma }=\left (q+\zeta -\mu \right).\ \)Since h(z) is a convex function in \(\mathbb {U\ }\)and
whenever (32) holds. Then \(f(z)\in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \)from Lemma 2. This completes the proof of (i). To prove (ii), we assume that (33) holds, then all the assumptions of Lemma 3 are satisfied for the above values of \(\widetilde {\beta },\ \widetilde {\gamma }\ \)and \(\widetilde {\alpha }=\frac {1-A}{1-B}.\ \)It follows that \(\mathcal {S}_{p,q}^{\lambda +1,\mu +1,\eta +1}\left (\zeta ;A,B\right) \subset \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta,\xi \right) \) where ξ(A,B) given by (34) is the best possible. □
Theorem 6
Assume that\(\ q\in \mathbb {N}_{0},\ p>q+\zeta \ {and}\ \)
(i) Suppose that \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon }f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast },\ \)then
(ii) Also, assuming that
then
where the bound
is the best possible.
Proof
Let \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ \)and
Since \(\left (\mathcal {M}_{0,z}^{\lambda,\mu,\eta,p}F_{p,\upsilon }f(z)\right)^{\left (q\right) }\neq 0\ \)for all \(z\in \mathbb {U}^{\ast },\ \)then H(z) is analytic in \(\mathbb {U}\) with H(0)=0 and H′(0)=1. Differentiating both sides of (41) logarithmically, we get
Differentiating both sides of (43) logarithmically, we get
Combining this identity together with \(f\left (z\right) \in \mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right),\ \)we obtain
We will use Lemma 2 for \(\widetilde {\beta }=\left (p-q-\zeta \right),\ \overline {\gamma }=\left (q+\zeta +\upsilon \right).\ \)Since h(z) is a convex function in \(\mathbb {U\ }\)and
whenever (38) holds. Then \(f(z)\in F_{p,\upsilon }\left (\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \right) \ \)from Lemma 2. This proves (i). To prove (ii), we assume that (39) holds, then all the assumptions of Lemma 3 are satisfied for \(\widetilde {\beta },\ \overline {\gamma }\ \) which stated above and \(\widetilde {\alpha }=\frac {1-A}{1-B}.\ \)It follows that
where τ(A,B) given by (40) is the best possible □
Conclusion
In our present investigation, we have derived some subordination results of certain subclasses of multivalent analytic functions which are defined by a generalized fractional differintegral operator. We have also successfully considered inclusion relations for functions in the class \(\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right)\) and the images of these functions by the generalized Bernardi-Libera-Livingston integral operator.
Abbreviations
- \(\mathcal {A}(p)\) :
-
The class of analytic and multivalent functions in the open unit disc \(\mathbb{U}=\{z\in \mathbb{C}|z| <1\}\)
- ∗:
-
Convolution of two power series
- ≺:
-
Subordination of two analytic functions in \(\mathbb {U\ }\)
- 2 F 1(a,b;c;z)(c≠0,−1,−2,…):
-
The well-known (Gaussian) hypergeometric function
- \(J_{0,z}^{\lambda,\mu,\eta,p}f(z)\) :
-
The generalized fractional derivative operator for \(f\in \mathcal {A}(p)\)
- F p, υ :
-
The generalized Bernardi-Libera-Livingston integeral operator
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Acknowledgements
The author would like to express her sincere thanks to Hillal Elshehabey from Mathematics department, Faculty of Science, South Valley university, for his help in plotting the function h(z), as well as the referees for their valuable suggestions and comments, which enhanced the paper.
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Zayed, H.M. Subordination and inclusion theorems for higher order derivatives of a generalized fractional differintegral operator. J Egypt Math Soc 27, 17 (2019). https://doi.org/10.1186/s42787-019-0020-2
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DOI: https://doi.org/10.1186/s42787-019-0020-2
Keywords
- Differential subordination
- Multivalent functions
- Higher order derivatives
- Generalized fractional differintegral operator