1 Introduction

A continuous function \(f=u+iv\) is a complex-valued harmonic function in a domain \(D {\subset } \mathbb {C}\) if both u and v are real harmonic in D. In any simply connected domain D, we write \(f=h+ \overline{g}\), where h and g are analytic in D. A necessary and sufficient condition for f to be locally one-to-one and orientation preserving in D is that \(|g^{\prime }(z)|<|h^{\prime }(z)|\) for \({z}\in D\) (see Clunie and Sheil-Small [4]). Functions that are harmonic and univalent in \({\mathbb {D}}=\{z:\left| {z}\right| >{1}\}\) are investigated by Hengartner and Schober [7]. In particular, it was shown in [7] that a complex-valued, harmonic, orientation preserving univalent mapping f, defined on \(\mathbb {D}\) and satisfying \(f(\infty )=\infty \), must admit the representation

$$\begin{aligned} f(z)=h(z)+{\overline{g(z)}}+A\log |z| \end{aligned}$$
(1)

where \(h(z)={\alpha }z+\sum _{n=0}^{\infty }a_{n}z^{-n}\), \(g(z)={\beta } z+\sum _{n=0}^{\infty }b_{n}z^{-n}\), \(0\ {\le }\ |\beta |<|\alpha |\), and \({ \omega }={\overline{f}}_{\overline{z}}/f_{z}\) is analytic and satisfies \(|{ \omega }(z)|<1\) for \({z}\ \in \ {\mathbb {D}}\). We remove the logarithmic singularity in (1) by letting \(A=0\) and also let \({\alpha }=1\) and \({ \beta }=0\) and focus on the family \({\Sigma }_{\mathcal {H}}\) of meromorphic harmonic orientation preserving univalent mappings of the form

$$\begin{aligned} f(z)=h(z)+\overline{g(z)} \end{aligned}$$

where

$$\begin{aligned} h(z)=z+{\sum }_{n=1}^{\infty }a_{n}z^{-n}\ ,\ \ g(z)={\sum }_{n=1}^{\infty }b_{n}z^{-n}\ . \end{aligned}$$
(2)

We say that a function \(f\in \Sigma _{\mathcal {H}}\) is harmonic starlike in \( \mathbb {D}\) if

$$\begin{aligned} \frac{\partial }{\partial t}\left( \arg f\left( re^{it}\right) \right) {\ge } \ 0 \end{aligned}$$

or

$$\begin{aligned} \mathrm {Re}\,\frac{D_{\mathcal {H}}f\left( z\right) }{f\left( z\right) }\ge { 0} \end{aligned}$$

where \(z = re^{it} {\in } {\mathbb {D}}\), \(0 {\le } t {\le } 2\pi \), and

$$\begin{aligned} D_{\mathcal {H}}f(z)=zh^{\prime }(z)-{\overline{zg^{\prime }(z)}} =z-\sum _{n=1}^{\infty }n\left( a_{n}z^{-n}-{\overline{b_{n}z^{-n}}}\right) . \end{aligned}$$

For \(l=1,2\) and functions \(f_{l}\in \Sigma _{\mathcal {H}}\) of the form

$$\begin{aligned} f_{l}(z)=z+\sum \limits _{n=1}^{\infty }\left( a_{l,n}z^{-n}+\overline{ b_{l,n}z^{-n}}\right) \end{aligned}$$
(3)

we definethe convolution of \(f_{1}\) and \(f_{2}\) by

$$\begin{aligned} \left( f_{1}*f_{2}\right) \left( z\right) =f_{1}(z)*f_{2}(z)=z+\sum \limits _{n=1}^{\infty }\left( a_{1,n}a_{2,n}z^{-n}+\overline{ b_{1,n}b_{2,n}z^{-n}}\right) . \end{aligned}$$

For \(m\in \mathbb {N}_{0}:=\left\{ 0\right\} {\cup } {\mathbb {N}}=\left\{ 1,2,...\right\} \) and \(f=h+{\overline{g}} {\in } {\Sigma }_{\mathcal {H}}\) we define the linear operator \(D_{\mathcal {H}}^{m}:{\Sigma }_{\mathcal {H} }\rightarrow {\Sigma }_{\mathcal {H}}\) by

$$\begin{aligned} D_{\mathcal {H}}^{m}:=D_{\mathcal {H}}^{m}f(z)= & {} \mathcal {D}^{m}h(z)+\left( -1\right) ^{m}\overline{\mathcal {D}^{m}g(z)} \\= & {} z+\sum \limits _{n=1}^{\infty }m_{n}\left\{ a_{n}z^{-n}+\left( -1\right) ^{m}\overline{b_{n}z^{-n}}\right\} \end{aligned}$$

where

$$\begin{aligned} \mathcal {D}^{m}h(z)= & {} h(z)*\left( z+\frac{\left( -1\right) ^{m}}{z(1-\frac{ 1}{z})^{m+1}}\right) =h(z)*\left( z+\frac{\left( -z\right) ^{m}}{ (z-1)^{m+1}}\right) ,\\ m_{1}=1, m_{n}= & {} \frac{\left( m+1\right) \cdot \ldots \cdot (m+n-1)}{\left( n-1\right) !}; \left( n=2,3,...\right) . \end{aligned}$$

We note that \(D_{\mathcal {H}}^{0}f=f\) and \(D_{\mathcal {H}}^{1}f=D_{\mathcal {H}}f\).

In 2008 Mauir [13] considered a weak subordination for complex-valued harmonic functions defined in the open unit disk \(\Delta :=\{z:\left| {z} \right| <{1}\}\). In [8] Jahangiri (see also [10]) investigated the classes of harmonic meromorphic starlike and convex functions of order \( \gamma .\) To obtain some generalizations of these classes we introduce definition of weak subordination for complex-valued functions in \(\mathbb {D} .\)

A complex-valued function f in\(\ \mathbb {D}\) is said to be weakly subordinate to a complex-valued function F in\(\ \mathbb {D}\), and we write \(f(z)\preceq F(z)\) (or simply \(f\preceq F\)), if \(f(\infty )=F(\infty )\) and \( f(\mathbb {D})\subset F(\mathbb {D}).\)

If \(f\preceq F\) and F is univalent in \(\mathbb {D}\), then we can consider the function \(\omega \left( z\right) =F^{-1}\left( f\left( z\right) \right) , z\in \mathbb {D}\) which maps \(\mathbb {D}\) into oneself with \( \omega (\infty )=\infty .\) Conversely, if \(\omega \left( z\right) =F^{-1}\left( f\left( z\right) \right) , z\in \mathbb {D},\) maps \(\mathbb {D}\) into oneself with \(\omega (\infty )=\infty \), then \(f\preceq F.\) Thus, we have the following equivalence.

Lemma 1

A complex-valued function f in \( \mathbb {D}\) is weakly subordinate to a function complex-valued function F in \( \mathbb {D}\) if and only if there exists a complex-valued function \(\omega \) which maps \(\mathbb {D}\) into oneself with \(\omega (\infty )=\infty \) such that \(f(z)=F(\omega (z)), z\in \mathbb {D}.\)

The following two subclasses of \({\Sigma }_{\mathcal {H}}\) are the main focus of this paper.

For \(-B\le A<B\le 1\) let \(\Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) be the class of functions \(f\in \Sigma _{\mathcal {H}}\) so that

$$\begin{aligned} \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f(z)\right) }{D_{\mathcal {H} }^{m}f\left( z\right) }\preceq \frac{A+z}{B+z} \end{aligned}$$

and \({\mathbb {V}}_{\mathcal {H}}^{m}\left( A,B\right) \) be the class of functions \(f {\in } \Sigma _{\mathcal {H}}\) so that

$$\begin{aligned} \frac{D_{\mathcal {H}}^{m}f\left( z\right) }{z}\preceq \frac{A+z}{B+z}. \end{aligned}$$

In particular, for \(m=0,m=1,\) we obtain classes studied in [5]. The classes \(\Sigma _{\mathcal {H}}^{*}(\gamma ):=\Sigma _{\mathcal {H} }^{1}(2\gamma -1,1) \)and\( \Sigma _{\mathcal {H}}^{c}(\gamma ):=\Sigma _{ \mathcal {H}}^{1}(2\gamma -1,1)\) are investigated by Jahangiri [8]. The classes \(\Sigma _{\mathcal {H}}^{*}:=\Sigma _{\mathcal {H}}^{*}(0)\) and \(\Sigma _{\mathcal {H}}^{c}:=\Sigma _{\mathcal {H}}^{c}(0)\) are the classes of functions \(f\in \Sigma _{\mathcal {H}}\left( k\right) \) which are starlike in \({\mathbb {U}}\left( r\right) \) or convex in \({\mathbb {U}}\left( r\right) ,\) respectively, for all \(r>1\) (see [10]).

We further let \(\mathcal {T}_{\eta }\) be the class of functions \(f=h+ \overline{g}\in {\Sigma }_{\mathcal {H}}\) with varying coefficients (e.g. see [9]) for which there exists a real number \(\eta \) so that

$$\begin{aligned} f(z)=h(z)+\overline{g(z)}=z+\sum \limits _{n=2}^{\infty }e^{i\left( n+1\right) \eta }\left| a_{n}\right| z^{-n}+\left( -1\right) ^{m+1}\overline{ \sum \limits _{n=2}^{\infty }e^{i\left( n-1\right) \eta }\left| b_{n}\right| {z}^{-n}} \end{aligned}$$
(4)

and \(\Sigma _{\eta }^{m}(A,B):=\mathcal {T}_{\eta }\cap \Sigma _{\mathcal {H} }^{m}\left( A,B\right) \) and \(V_{\eta }^{m}(A,B):=\mathcal {T}_{\eta }\cap V_{ \mathcal {H}}^{m}\left( A,B\right) .\)

2 Dual sets and coefficient bounds

Here we make use of the “duality principle” defined by Ruscheweyh ([14] , Chapter 1). For the set \(\mathcal {V}\ {\subset } {\Sigma }_{\mathcal {H}}\) we define the dual set of \(\mathcal {V}\) by

$$\begin{aligned} \mathcal {V}^{*}=\left\{ {f_2}\in \Sigma _{\mathcal {H}} | {\forall } { f_1}{ \in } {\mathcal {V}} : \left( { f_1}*{f_2}\right) \left( z\right) \ne 0 ;{ z\in \mathbb {D} }\right\} . \end{aligned}$$

Consequently, the second dual of \(\mathcal {V}\) or the dual of \({\mathcal {V}} ^{*}\) is defined as

$$\begin{aligned} \mathcal {V}^{{*}{*}}={(\mathcal {V}^{*})}^{*}=\left\{ {f_3}\in \Sigma _{\mathcal {H}} | {\forall } {f_2}{\in } {\mathcal {V}}^{*} : \left( { f_2}*{f_3}\right) \left( z\right) \ne 0 ;{ z\in \mathbb {D} }\right\} . \end{aligned}$$

This duality principle indicates that under fairly weak conditions on \( \mathcal {V}\) which is a subset of \({\Sigma }_{\mathcal {H}}\), many linear and other extremal problems in the second dual of \(\mathcal {V}\) are solved in \( \mathcal {V}\). This is a very useful tool since in many cases of interest (such as convex, starlike or close-to-convex harmonic univalent functions), the set \({\mathcal {V}}^{{*}{*}}\) is much larger than the set \(\mathcal { V}\). This, on its own right, would be a separate endeavor and research topic that can be explored further which is not the focus of the present paper. The following theorem presents a duality condition for the set \({\Sigma }_{ \mathcal {H}}^m(A,B)\).

Theorem 1

$$\begin{aligned} \Sigma _{\mathcal {H}}^{m}\left( A,B\right) =\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*},\ \end{aligned}$$

where

$$\begin{aligned} \psi _{\xi }\left( z\right)&:=\left( B-A\right) z+(-1)^{m+1} \frac{\left( B+A+2\xi \right) z-\left( 1+m\right) \xi +\left( \lambda B+A\right) }{ (1-z)^{m+2}}z^m \\&\quad +\left( -1\right) ^{m+1}\left( 2\xi +B+A\right) \overline{z}\\&\quad + \frac{\left( B-A\right) \overline{z}+\left( 1-m\right) \xi -\left( \lambda B-A\right) }{ (1-\overline{z})^{m+2}}{\overline{z}}^{m}; \left( z\in \mathbb {D} \right) . \end{aligned}$$

Proof

Let \(f\in \Sigma _{\mathcal {H}}\). Then \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) if and only if

$$\begin{aligned} \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) }{ D_{ \mathcal {H}}^{m}f\left( z\right) }\ne \frac{A+\xi }{B+\xi } ; \left( z\in \mathbb {D},~\left| \xi \right| =1\right) . \end{aligned}$$
(5)

Since

$$\begin{aligned} D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}h\right) \left( z\right)= & {} z\left( D_{\mathcal {H}}^{m}h\right) ^{\prime }\left( z\right) =h\left( z\right) *\left\{ z\left( z+\frac{\left( -z\right) ^{m}}{(z-1)^{m+1}}\right) ^{\prime }\right\} \\= & {} h\left( z\right) *\left( z-\frac{\left( z-m\right) \left( -z\right) ^{m}}{(z-1)^{m+2}}\right) , \end{aligned}$$

the above inequality (5) yields

$$\begin{aligned}&\left( B+\xi \right) D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -\left( A+\xi \right) D_{\mathcal {H}}^{m}f\left( z\right) \\&=\left( B+\xi \right) D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}h\right) \left( z\right) -\left( A+\xi \right) D_{\mathcal {H}}^{m}h\left( z\right) \\&\quad -\left( -1\right) ^{m}\left[ \left( B+\xi \right) \overline{D_{\mathcal {H} }\left( D_{\mathcal {H}}^{m}g\right) \left( z\right) }+\left( A+\xi \right) \overline{D_{\mathcal {H}}^{m}h\left( z\right) }\right] \\&=h\left( z\right) *\left( \left( B+\xi \right) \left( z-\frac{\left( z-m\right) \left( -z\right) ^{m}}{(z-1)^{m+2}}\right) -\left( A+\xi \right) \left( z+\frac{\left( -z\right) ^{m}}{(z-1)^{m+1}}\right) \right) \\&\quad -\left( -1\right) ^{m}\overline{g\left( z\right) }*\left( \left( B+\xi \right) \left( \overline{z}-\frac{\left( \overline{z}-m\right) \left( \overline{z}\right) ^{m}}{(\overline{z}-1)^{m+2}}\right) \right. \\&\quad \left. +\left( A+\xi \right) \left( \overline{z}+\frac{\left( -\overline{z}\right) ^{m}}{( \overline{z}-1)^{m+1}}\right) \right) \\&=f\left( z\right) *\psi _{\xi }\left( z\right) \ne 0. \end{aligned}$$

Thus, \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) if and only if \( f\left( z\right) *\psi _{\xi }\left( z\right) \ne 0\) i.e. \( \Sigma _{\mathcal {H}}^{m}\left( A,B\right) =\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}\). \(\square \)

A similar argument can be used to obtain a duality condition for the set \({ \mathbb {V}}_{\mathcal {H}}^m(A,B)\).

Theorem 2

$$\begin{aligned} {\mathbb {V}}_{\mathcal {H}}^{m}\left( A,B\right) =\left\{ \delta _{\xi }:\left| \xi \right| =1\right\} ^{*}, \end{aligned}$$

where

$$\begin{aligned} \delta _{\xi }\left( z\right)&:&=\left( B-A\right) z+\left( -1\right) ^{m+1} \frac{\left( 1+B\xi \right) z^{m}}{(z-1)^{m+1}} \\&+ \left( -1\right) ^{m}\left( B+\xi \right) \overline{z}-\frac{\left( B+\xi \right) \overline{z}^{m}}{(\overline{z}-1)^{m+1}}\left( z\in \mathbb {D}\right) . \end{aligned}$$

Next we determine sufficient coefficient bounds for function in \(\Sigma _{\mathcal {H}}^{m}\left( A,B\right) \).

Theorem 3

Let \(f=h+\overline{g}\) be of the form (2) and \(-B\ {\le } \ A<B\ {\le }\ 1\). If

$$\begin{aligned} \sum \limits _{n=1}^{\infty }m_{n}\left\{ \left( n(1+B)+(1+A)\right) \left| a_{n}\right| +\left( n(1+B)-(1+A)\right) \left| b_{n}\right| \right\} \le B-A, \end{aligned}$$

then f is univalent and orientation preserving in \(\mathbb {D}\) and \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \).

Proof

The univalency and orientation preserving of the function f follows by a result of Jahangiri and Silverman ([10], Theorem 1) since

$$\begin{aligned} n(B-A){\le } m_n(n(1+B)-(1+A)) {\le } m_n(n(1+B)+(1+A)). \end{aligned}$$

Therefore, by Lemma 1 \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) if and only if there exists a complex-valued function \(\omega \) where \( \omega (\infty )=\infty \), \(\left| \omega (z)\right| >1\) and \(z\in \mathbb {D}\) such that

$$\begin{aligned} \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) }{D_{ \mathcal {H}}^{m}f\left( z\right) }=\frac{A+\omega (z)}{B+\omega (z)} \end{aligned}$$

or equivalently

$$\begin{aligned} \left| \frac{D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H}}^{m}f\left( z\right) }{BD_{\mathcal {H}}\left( D_{ \mathcal {H}}^{m}f\right) \left( z\right) -A\left( D_{\mathcal {H}}^{m}f\left( z\right) \right) \left( z\right) }\right| <1 . \end{aligned}$$
(6)

Thus for \(z\in \mathbb {D}\) it suffices to show that

$$\begin{aligned} \left| D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H}}^{m}f\left( z\right) \right| -\left| BD_{\mathcal {H} }\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H} }^{m}f\left( z\right) \right| <0. \end{aligned}$$

Indeed, letting \(\left| z\right| =r\,~(r>1)\) we have

$$\begin{aligned}&{\left| D_{\mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H}}^{m}f\left( z\right) \right| -\left| BD_{ \mathcal {H}}\left( D_{\mathcal {H}}^{m}f\right) \left( z\right) -D_{\mathcal {H }}^{m}f\left( z\right) \right| } \\&\quad =\left| \sum \limits _{n=1}^{\infty }\left( n+1\right) m_{n}a_{n}z^{-n}-\left( -1\right) ^{m}\sum \limits _{n=1}^{\infty }\left( n-1\right) m_{n}\overline{b_{n}}\overline{z}^{-n}\right| \\&\qquad -\left| \left( B-A\right) z+\sum \limits _{n=1}^{\infty }\left( Bn+A\right) m_{n}a_{n}z^{-n}+\left( -1\right) ^{m}\sum \limits _{n=1}^{\infty }\left( Bn-A\right) m_{n}\overline{b_{n}}\overline{z}^{-n}\right| \\&\quad \le \sum \limits _{n=1}^{\infty }\left( n+1\right) m_{n}\left| a_{n}\right| r^{-n}+\sum \limits _{n=1}^{\infty }\left( n-1\right) m_{n}\left| b_{n}\right| r^{-n}-\left( B-A\right) r \\&\qquad +\sum \limits _{n=1}^{\infty }\left( Bn+A\right) m_{n}\left| a_{n}\right| r^{-n}+\sum \limits _{n=1}^{\infty }\left( Bn-A\right) m_{n}\left| b_{n}\right| r^{-n} \\&\quad \le r\left\{ \sum \limits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) r^{-n-1}-\left( B-A\right) \right\} <0 \end{aligned}$$

where \({\alpha }_n=m_n(n(1+B)+(1+A))\) and \({\beta }_n=m_n(n(1+B)-(1+A)))\). Thus, according to the hypothesis of the theorem, \(f\in \Sigma _{\mathcal {H} }^{m}\left( A,B\right) \). \(\square \)

The sufficient coefficient bound given in Theorem 3 is also necessary for functions to be in the class \(\Sigma _{\eta }^{m}(A,B)\) as stated in the following theorem.

Theorem 4

Let \(f=h+\overline{g}\) be of the form (4) and \(-B\ {\le }\ A<B\ {\le }\ 1\). Then \(f\in \Sigma _{\eta }^{m}(A,B)\) if and only if

$$\begin{aligned} \sum \limits _{n=1}^{\infty }m_{n}\left\{ \left( n(1+B)+(1+A)\right) \left| a_{n}\right| +\left( n(1+B)-(1+A)\right) \left| b_{n}\right| \right\} \le B-A. \end{aligned}$$

Proof

The “if” part follows from Theorem 3. For the “only-if” part, assume that \(f\in \Sigma _{\eta }^{m}(A,B),\) then by (6) we have

$$\begin{aligned} \left| \frac{\sum \limits _{n=1}^{\infty }m_{n}\left\{ \left( n+1\right) a_{n}z^{-n}-\left( -1\right) ^{m}\left( n-1\right) \overline{b_{n}}\overline{ z}^{-n}\right\} }{\left( B-A\right) z-\sum \limits _{n=1}^{\infty }m_{n}\left\{ \left( Bn+A\right) a_{n}z^{-n}-\left( -1\right) ^{m}\left( Bn-A\right) \overline{b_{n}}\overline{z}^{-n}\right\} }\right| <1 \,\,(z\in \mathbb {D}). \end{aligned}$$

Therefore, by (4) for \(z=re^{i\eta }~(r>1),\) we obtain

$$\begin{aligned} \frac{\sum \nolimits _{n=1}^{\infty }\left\{ \left( n+1\right) \left| m_{n}\right| \left| a_{n}\right| +\left( n-1\right) \left| m_{n}\right| \left| b_{n}\right| \right\} r^{-n-1}}{\left( B-A\right) -\sum \nolimits _{n=1}^{\infty }\left\{ \left( Bn+A\right) \left| m_{n}\right| \left| a_{n}\right| +\left( Bn-A\right) \left| m_{n}\right| \left| b_{n}\right| \right\} r^{-n-1}}<1. \end{aligned}$$
(7)

It is clear that the denominator of the ratio in the inequality (7) cannot vanish for \(r>1.\) Moreover, it is positive as \(r{\rightarrow }\infty \) and consequently for \(r>1.\) Thus, we must have

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) r^{-n-1}<B-A (r>1) \end{aligned}$$
(8)

where \({\alpha }_n=m_n(n(1+B)+(1+A))\) and \({\beta }_n=m_n(n(1+B)-(1+A)))\). The sequence of partial sums \(\left\{ S_{n}\right\} \) associated with the series \(\sum \nolimits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \) is a non-decreasing sequence. Moreover, by (8), it is bounded above by \(B-A.\) Hence, the sequence\(\ \left\{ S_{n}\right\} \) is convergent and

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) =\lim _{n\rightarrow \infty }\{S_{n}\} {\le } B-A. \end{aligned}$$

\(\square \)

A similar argument can be used to prove the following theorem.

Theorem 5

Let \(f=h+\overline{g}\) be of the form (2) and \(-B\ {\le } \ A<B\ {\le }\ 1\). Then \(f\in {\mathbb {V}}_{\eta }^{m}(A,B)\) if and only if

$$\begin{aligned} \sum \limits _{n=1}^{\infty }m_{n}\left( \left| a_{n}\right| +\left| b_{n}\right| \right) \le \frac{B-A}{1+B}. \end{aligned}$$

As a consequence of Theorems 4 and 5 we have the following corollary.

Corollary 1

For \(-B\ {\le }\ A<B\ {\le }\ 1\) let \(\ a=\frac{1+A}{1+B}\),

$$\begin{aligned} \phi \left( z\right)= & {} z+\sum \limits _{n=1}^{\infty }\left( \frac{1}{n-a} z^{-n}+\frac{1}{n+a}\overline{{z}^{-n}}\right) \left( z\in \mathbb {D} \right) , \\ \omega \left( z\right)= & {} z+\sum \limits _{n=1}^{\infty }\left( \left( n-a\right) z^{-n}+\left( n+a\right) \overline{{z}^{-n}}\right) \left( z\in \mathbb {D}\right) . \end{aligned}$$

Then

$$\begin{aligned} f\in & {} {\mathbb {V}}_{\eta }^{m}(A,B)\Leftrightarrow f*\phi \in \Sigma _{\eta }^{m}(A,B), \\ f\in & {} \Sigma _{\eta }^{m}(A,B)\Leftrightarrow f*\omega \in {\mathbb {V}} _{\eta }^{m}(A,B). \end{aligned}$$

3 Extreme points

The Krein-Milman theorem (see [11]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.

Lemma 2

Let \(\mathcal {F}\) be a non-empty compact convex subset of the set \({\Sigma }_{\mathcal {H}}\) and \(\mathcal {J}:{\Sigma }_{\mathcal {H} }\rightarrow \mathbb {R}\) be a real-valued, continuous and convex functional on \(\mathcal {F}.\) Then

$$\begin{aligned} \max \left\{ \mathcal {J}(f):f\in \mathcal {F}\right\} =\max \left\{ \mathcal {J }(f):f\in E\mathcal {F}\right\} , \end{aligned}$$

where \(E\mathcal {F}\) denotes the set of extreme points of \(\mathcal {F}.\)

Since \({\Sigma }_{\mathcal {H}}\) is a complete metric space, Montel’s theorem [12] implies the following lemma.

Lemma 3

A set \(\mathcal {F}\subset {\Sigma }_{\mathcal {H}} \) is compact if and only if \(\mathcal {F}\) is closed and locally uniformly bounded.

Now we are equipped to state and prove the two main theorems in this section.

Theorem 6

The set \(\Sigma _{\eta }^{m}(A,B)\) is a convex and compact subset of \({\Sigma }_{\mathcal {H}} \).

Proof

For \({l}=1,2\) let \(f_{l}\in \Sigma _{\eta }^{m}(A,B)\) be functions of the form (3),\({\;0\le \gamma \le 1.}\) Since

$$\begin{aligned} \gamma f_{1}(z)+(1-\gamma )f_{2}\left( z\right)= & {} z+\sum \limits _{n=1}^{\infty }\left\{ \left( \gamma a_{1,n}+\left( 1-\gamma \right) a_{2,n}\right) z^{-n}\right. \\&\left. + \overline{\left( \gamma b_{1,n}+\left( 1-\gamma \right) b_{2,n}\right) z^{-n} }\right\} , \end{aligned}$$

and since for

$$\begin{aligned} {\alpha }_{n}=m_{n}(n(1+B)+(1+A)), {\beta }_{n}=m_{n}(n(1+B)-(1+A))) \end{aligned}$$
(9)

(by Theorem 4), we have

$$\begin{aligned}&\sum \limits _{n=1}^{\infty }\left\{ \alpha _{n}\left| \gamma a_{1,n}+\left( 1-\gamma \right) a_{2,n}\right| +\beta _{n}\left| \gamma b_{1,n}+\left( 1-\gamma \right) b_{2,n}\right| \right\} \\&\le \gamma \sum \limits _{n=1}^{\infty }\left\{ \alpha _{n}\left| a_{1,n}\right| +\beta _{n}\left| b_{1,n}\right| \right\} +\left( 1-\gamma \right) \sum \limits _{n=1}^{\infty }\left\{ \alpha _{n}\left| a_{2,n}\right| +\beta _{n}\left| b_{2,n}\right| \right\} \\&\le \gamma \left( B-A\right) +\left( 1-\gamma \right) \left( B-A\right) =B-A, \end{aligned}$$

the function \({\phi }=\gamma f_{1}+(1-\gamma )f_{2}\) belongs to the class \( \Sigma _{\eta }^{m}(A,B)\). Hence, the set \(\Sigma _{\eta }^{m}(A,B)\) is convex. Furthermore, for \(f\in \Sigma _{\eta }^{m}(A,B)\) and \(\left| z\right| = r>1\) we have

$$\begin{aligned} \left| f(z)\right| \le r+\sum \limits _{n=1}^{\infty }\left( \left| a_{n}\right| +\left| b_{n}\right| \right) r^{-n}\le r+\sum \limits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \le r+\left( B-A\right) . \end{aligned}$$

Thus, the class \(\Sigma _{\eta }^{m}(A,B)\) is locally uniformly bounded. Now, by Lemma 3, we only need to show that it is closed i.e. if \(f_{l}\in \Sigma _{\eta }^{m}(A,B) \) and \(f_{l}\rightarrow f\) then \(f\in \Sigma _{\eta }^{m}(A,B).\) Let \(f_{l}\) and f be given by (3) and (4), respectively. Using Theorem 4 we have

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{l,n}\right| +\beta _{n}\left| b_{l,n}\right| \right) \le B-A~\left( l\in \mathbf {\mathbb {N}}\right) .\, \end{aligned}$$
(10)

Since \(f_{l}\rightarrow f,\) we conclude that \(a_{l,n}\rightarrow a_{n}\) and \( b_{l,n}\rightarrow b_{n}\) as \(l\rightarrow \infty \left( n\in \mathbf { \mathbb {N}}\right) \). The sequence of partial sums \(\left\{ S_{n}\right\} \) associated with the series \(\sum \nolimits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \) is a non-decreasing sequence. Moreover, by (10), it is bounded above by \(B-A.\) Therefore, the sequence\(\left\{ S_{n}\right\} \) is convergent and

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) =\lim _{n\rightarrow \infty }S_{n}\le B-A. \end{aligned}$$

This implies that \(f\in \Sigma _{\eta }^{m}(A,B)\) which completes the proof. \(\square \)

In the following theorem we determine the extreme points of \(\Sigma _{\eta }^{m}(A,B)\).

Theorem 7

$$\begin{aligned} E\Sigma _{\eta }^{m}(A,B)=\left\{ h_{n}:\;n\in \mathbb {N}_{0}\right\} \cup \left\{ g_{n}:\;n\in \mathbb {N}\right\} , \end{aligned}$$

where

$$\begin{aligned} h_{0}(z)&=z,\ h_{n}(z)=z+\frac{\left( B-A\right) {e^{i\left( 1+n\right) \eta }}}{m_{n}(n(1+B)+(1+A))}z^{-n},\; \nonumber \\ g_{n}(z)&=z+\frac{\left( -1\right) ^{m+1}\left( B-A\right) {e^{i\left( 1-n\right) \eta }}}{m_{n}(n(1+B)-(1+A)))}\overline{{z}^{-n}}\ \ \ \left( z\in \mathbb {D}\right) . \end{aligned}$$
(11)

Proof

Suppose that \(0<\gamma <1\) and \(g_{n}=\gamma f_{1}+\left( 1-\gamma \right) f_{2}\) where \(f_{1},f_{2}\in \Sigma _{\eta }^{m}(A,B)\) are functions of the form (3) and let \(\alpha _{n},\beta _{n}\) be defined by (9). Then \(\left| b_{1,n}\right| =\left| b_{2,n}\right| =\) \(\frac{ B-A}{\beta _{n}}\) and consequently \(a_{1,n}=a_{2,n}=0\) and \( b_{1,k}=b_{2,k}=0 \) for \(k\in \mathbf {\mathbb {N}} \diagdown \left\{ n\right\} .\) It follows that \(g_{n}=f_{1}=f_{2},\) and so \(g_{n}\in \Sigma _{\eta }^{m}(A,B)\). Similarly, we can verify that the functions \(h_{n}\) of the form (11) are the extreme points of the class \(\Sigma _{\eta }^{m}(A,B).\) Now, suppose that the function f belongs to the set \( E\Sigma _{\eta }^{m}(A,B)\) and f is not of the form (11). Then there exists \(s\in \mathbf {\mathbb {N}}\) such that

$$\begin{aligned} 0<\left| a_{s}\right|<\frac{B-A}{{\alpha }_{s}}\text { \ or \ } 0<\left| b_{s}\right| <\frac{B-A}{{\beta }_{s}}. \end{aligned}$$

If \(0<\left| a_{s}\right| <\frac{B-A}{{\alpha }_{s}}\), then setting

$$\begin{aligned} \gamma =\frac{{\alpha }_{s}\left| a_{s}\right| }{B-A}\ and ~\varphi = \frac{1}{1-\gamma }\left( f-\gamma h_{s}\right) , \end{aligned}$$

we have that \(h_{s}\ne \varphi \) and so \(f=\gamma h_{s}+\left( 1-\gamma \right) \varphi ,\ 0<\gamma <1\). Thus, \(f\notin E\Sigma _{\eta }^{m}(A,B).\)

Similarly, if \(0<\left| b_{s}\right| <\frac{B-A}{\beta _{n}}\), then setting

$$\begin{aligned} \gamma =\frac{{\beta }_{s}\left| b_{s}\right| }{B-A},~\phi =\frac{1}{ 1-\gamma }\left( f-\gamma g_{s}\right) , \end{aligned}$$

we have that \(g_{s}\ne \phi \) and so \(f=\gamma g_{s}+\left( 1-\gamma \right) \phi ,\ 0<\gamma <1\).

It follows that \(f\notin E\Sigma _{\eta }^{m}(A,B)\) and so the proof is completed. \(\square \)

4 Applications

It is clear that if the class

$$\begin{aligned} \mathcal {F}=\left\{ f_{n}\in {\Sigma }_{\mathcal {H}}:\;n\in \mathbf {\mathbb {N }}\right\} , \end{aligned}$$

is locally uniformly bounded, then

$$\begin{aligned} \overline{co}\mathcal {F}=\left\{ \sum _{n=1}^{\infty }\gamma _{n}f_{n}:~\sum _{n=1}^{\infty }\gamma _{n}=1,~\gamma _{n}\ge 0~\left( n\in \mathbf {\mathbb {N}}\right) \right\} . \end{aligned}$$

Thus, by Theorem 7, we have the following corollary.

Corollary 2

$$\begin{aligned} \Sigma _{\eta }^{m}(A,B):=\left\{ \sum _{n=0}^{\infty }\gamma _{n}h_{n}+\sum _{n=1}^{\infty }\delta _{n}g_{n} : ~\sum _{n=0}^{\infty }\gamma _{n}+\sum _{n=1}^{\infty }\delta _{n} =1,\ \gamma _{n}\ {\ge }\ 0,\ \delta _{n}\ge 0 \right\} \end{aligned}$$

where \(h_{n}\) and \(g_{n}\) are given by (11).

For \(f\ {\in }\ {\Sigma }_{\mathcal {H}}\), \(\gamma \ge 1\) and \(r>1\) the real-valued functional

$$\begin{aligned} {\mathcal {J}}\left( f\right) =\left( \frac{1}{2\pi }\int \limits _{0}^{2\pi } \left| f\left( re^{i\theta }\right) \right| ^{\gamma }d\theta \right) ^{1/\gamma } \end{aligned}$$

is continuous and convex on \({\Sigma }_{\mathcal {H}}\).

Moreover, for \(f\ {\in }\ {\Sigma }_{\mathcal {H}}\), \(z\ {\in }\ {\mathbb {D}}\) and each fixed \(n\ {\in }\ {\mathbb {N}}\) the real-valued functionals \({ \mathcal {J}}_1\left( f\right) =a_{n}\), \({\mathcal {J}}_2\left( f\right) =b_{n} \), \({\mathcal {J}}_3\left( f\right) =\left| f\left( z\right) \right| \) and \({\mathcal {J}}_4\left( f\right) =\left| D_{\mathcal {H} }f\left( z\right) \right| \) are also continuous and convex on \({\Sigma }_{ \mathcal {H}}\).

These in conjunction with Lemma 2 yield the following corollaries.

Corollary 3

Let \(f\in \Sigma _{\eta }^{m}(A,B)\). Then

$$\begin{aligned} \left| a_{n}\right| \le {\frac{B-A}{m_{n}(n(1+B)+(1+A))} and} \left| b_{n}\right| \le {\frac{B-A}{m_{n}(n(1+B)-(1+A)))}.} \end{aligned}$$

The result is sharp and the functions \(h_{n}\) and \(g_{n}\) of the form (11) are the extremal functions.

Corollary 4

Let \(f\in \Sigma _{\eta }^{m}(A,B)\). Then

$$\begin{aligned} \,r-\frac{1}{r}\le \left| f(z)\right| \le r+\frac{1}{r} \end{aligned}$$

and

$$\begin{aligned} r-\frac{1}{r}\le \left| D_{\mathcal {H}}f(z)\right| \le r+\frac{1}{r }. \end{aligned}$$

The result is sharp and the extremal function is given by (11).

The following covering result follows from Corollary 4.

Corollary 5

If \(f\in \Sigma _{\eta }^{m}(A,B),\) then \(\mathbb {D}\left( 2\right) \subset f\left( \mathbb {D}\right) .\)

Corollary 6

Let \(r>1\) and \(\gamma \ge 1\). If \(f\in \Sigma _{\eta }^{m}(A,B),\) then

$$\begin{aligned} \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| f(re^{i\theta })\right| ^{\gamma }d\theta\le & {} \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| r- \frac{1}{re^{i\theta }}\right| ^{\gamma }d\theta , \\ \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| D_{\mathcal {H} }f(z)\right| ^{\gamma }d\theta\le & {} \frac{1}{2\pi }\int \limits _{0}^{2 \pi }\left| r-\frac{1}{re^{i\theta }}\right| ^{\gamma }d\theta . \end{aligned}$$

Corollary 7

The class \({\mathbb {V}}_{\eta }^{m}(A,B)\) is a convex compact subset of \({ \Sigma }_{\mathcal {H}} \). Moreover,

$$\begin{aligned} E{\mathbb {V}}_{\eta }^{m}(A,B)=\left\{ h_{n}:\;n\in \mathbb {N}_{0}\right\} \cup \left\{ g_{n}:\;n\in \mathbb {N}\right\} \end{aligned}$$

and

$$\begin{aligned} {\mathbb {V}}_{\eta }^{m}(A,B)=\left\{ \sum _{n=0}^{\infty }\gamma _{n}h_{n}+\sum _{n=1}^{\infty }\delta _{n}g_{n} : \ {\sum } _{n=0}^{\infty }\gamma _{n}+{\sum }_{n=1}^{\infty }\delta _{n}=1, \ {\gamma _{n} }\ {\ge }\ 0,\ {\delta _{n}}\ {\ge } 0 \right\} \end{aligned}$$

where \(h_{0}(z)=z,\) and

$$\begin{aligned} h_{n}(z)= & {} z+\frac{\left( B-A\right) {e^{i\left( 1+n\right) \eta }}}{\left( 1+B\right) m_{n}}z^{-n},\nonumber \\ g_{n}(z)= & {} z+\left( -1\right) ^{m+1}\frac{\left( B-A\right) {e^{i\left( 1-n\right) \eta }}}{\left( 1+B\right) m_{n}}\overline{ {z}^{-n}}. \end{aligned}$$
(12)

Corollary 8

Let \(f\in {\mathbb {V}}_{\eta }^{m}(A,B)\) be a function of the form (4) . Then

$$\begin{aligned}&\left| a_{n}\right| \le \frac{B-A}{\left( 1+B\right) m_{n}}{,\ } \left| b_{n}\right| \le \frac{B-A}{\left( 1+B\right) m_{n}} ~~\,\,(n\in \mathbb {N} ), \\&r- \frac{B-A}{\left( 1+B\right) r}\le \left| f(z)\right| \le r+ \frac{B-A}{\left( 1+B\right) r}, \\&r- \frac{B-A}{\left( 1+B\right) r}\le \left| D_{\mathcal {H} }f(z)\right| \le r+\frac{B-A}{\left( 1+B\right) r}, \\&\frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| f(re^{i\theta })\right| ^{\gamma }d\theta \le \frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| r-\frac{B-A}{\left( 1+B\right) re^{i\theta }}\right| ^{\gamma }d\theta , \\&\frac{1}{2\pi }\int \limits _{0}^{2\pi }\left| D_{\mathcal {H} }f(re^{i\theta })\right| ^{\gamma }d\theta \le \frac{1}{2\pi } \int \limits _{0}^{2\pi }\left| r-\frac{B-A}{\left( 1+B\right) re^{i\theta }}\right| ^{\gamma }d\theta . \end{aligned}$$

The results are sharp and the functions \(h_{n}\) and \(g_{n}\) of the form (12) are the extremal functions.

Corollary 9

If \(f\in {\mathbb {V}}_{\eta }^{m}(A,B),\) then \(\mathbb {D}\left( r\right) \subset f\left( \mathbb {D}\right) \) where

$$\begin{aligned} \,r=1+\frac{B-A}{1+B}. \end{aligned}$$

Remark 1

The classes \({\Sigma }_{\mathcal {H}}^{m}(A,B)\) and \({\mathbb {V}}_{\mathcal {H} }^{m}(A,B)\) for different values of m, A and B give various well-known as well as new classes of meromorphic harmonic univalent functions (see for example [2, 3, 5, 6, 8, 10]).