Ruscheweyh-type meromorphic harmonic functions

Abstract

In this paper, we study classes of meromorphic harmonic functions defined by Ruscheweyh derivatives. In addition to finding certain analytic criteria, we obtain radii of starlikeness and convexity, and some topological properties for the defined classes of functions. Some applications of these results are also given.

1 Introduction

A complex-valued function f is said to be harmonic in a domain $D\subset \mathbb{C}$ if it has continuous second-order partial derivatives in D that satisfy the Laplace equation

$$ \Delta f:=\frac{\partial ^{2}f}{\partial x^{2}}+ \frac{\partial ^{2}f}{\partial y^{2}}=0. $$

If $D={\mathbb{U}} ( r ) :={\{z\in \mathbb{C}: }0< \vert z \vert <{r\},}$ then we say that f is a meromorphic harmonic function in ${\mathbb{U}} ( r )$. We denote by $\mathcal{M}$ the class of all such function with the normalization $f ( 0 ) =\infty$

Let a function F be harmonic, orientation-preserving, and univalent in ${\mathbb{B}}:={\{z\in \mathbb{C}: } \vert {z} \vert >{1\}}$ with $F(\infty )=\infty$. Then, there exists $B\in {\mathbb{C}}$ and functions

$$ \varphi (z)=\alpha z+\sum_{n=1}^{\infty }a_{n}z^{-n},\qquad \psi (z)=\beta z+\sum_{n=1}^{\infty }b_{n}z^{-n}\quad \bigl( 0\leq \vert \alpha \vert < \vert \beta \vert , z\in \mathbb{B} \bigr) , $$

such that

$$ F ( z ) =\varphi ( z ) + \overline{\psi ( z ) }+B\log \vert z \vert , \quad ( z\in \mathbb{B} ) $$

where $\overline{F_{\overline{z}}}/F_{z}$ is analytic and bounded by 1 in $\mathbb{E}$ (see, Hengartner and Schober [10]).

Let $f\in \mathcal{M}$ be functions that are univalent and sense-preserving in ${\mathbb{U}}:={\mathbb{U}} ( 1 )$. Since the composition of an analytic and harmonic function is the harmonic function, the function $F=f\circ ( \frac{1}{z} )$ is orientation-preserving, harmonic, and univalent in ${\mathbb{E}}$ with $F(\infty )=\infty$. Thus, there exists $B\in {\mathbb{C}}$ and the functions $h ( z ) :=\varphi ( \frac{1}{z} )$, $g ( z ) :=\psi ( \frac{1}{z} )$ such that

$$ f ( z ) =h ( z ) + \overline{g ( z ) }-B\log \vert z \vert \quad ( z\in { \mathbb{U}} ) . $$

Let $k\in {\mathbb{N}}:={\mathbb{N}}_{1}$, where ${\mathbb{N}}_{m}:= \{ m,m+1,\ldots \}$. We denote by $\mathcal{M}_{\mathcal{H}} ( k )$ the class of functions $f\in \mathcal{M}$ of the form

$$ f=h+\overline{g},\qquad h ( z ) =\frac{1}{z}+\sum_{n=k}^{ \infty }a_{n}z^{n},\qquad g ( z ) =\sum_{n=k}^{\infty }b_{n}z^{n}\quad ( z\in {\mathbb{U}} ) , $$

(1)

which are sense-preserving and univalent in ${\mathbb{U}}$, and let $\mathcal{M}_{\mathcal{H}}:=\mathcal{M}_{\mathcal{H}} ( 1 )$.

Recently, classes of meromorphic harmonic functions were intensively studied (see for example [1–11]).

A function $f\in \mathcal{M}_{\mathcal{H}} ( k )$ is called meromorphic harmonic starlike in $\mathbb{U} ( r )$ if f maps $\partial \mathbb{U} ( r )$ onto a curve that is starlike with respect to the origin, i.e.,

$$ \frac{\partial }{\partial t} \bigl( \arg f \bigl( re^{it} \bigr) \bigr) < 0 \quad ( 0\leq t\leq 2\pi ) $$

(2)

or equivalently

$$ \mathrm{Re} \frac{\mathcal{D}_{\mathcal{H}}f ( z ) }{f ( z ) }< 0 \quad \bigl( \vert z \vert =r \bigr) , $$

where

$$ \mathcal{D}_{\mathcal{H}}f ( z ) :=-zh^{\prime } ( z ) +\overline{zg^{\prime } ( z ) } \quad ( z\in \mathbb{U} ) . $$

Let φ and Φ be complex-valued functions in ${ \mathbb{U}}$. If $\varphi ({\mathbb{U}})\subset \Phi ({\mathbb{U}})$, then we say that φ is weakly subordinate to Φ, and we write $\varphi (z)\preceq \Phi (z)$ (see Muir [16]).

For functions

$$ f_{l} ( z ) =\sum_{n=-1}^{\infty } \bigl( a_{l,n}z^{n}+\overline{b_{l,n}z^{n}} \bigr)\quad ( z\in {\mathbb{U}}, l=1,2, ) $$

we define the convolution of functions $f_{1}$ and $f_{2}$ by

$$ ( f_{1}\ast f_{2} ) ( z ) =\sum _{n=-1}^{ \infty } \bigl( a_{1,n}a_{2,n}z^{n}+ \overline{b_{1,n}b_{2,n}z^{n}} \bigr) \quad ( z\in { \mathbb{U}} ) . $$

In [17] Ruscheweyh introduced an operator $\mathcal{D}^{\lambda }$ defined on the class of analytic functions by

$$ \mathcal{D}^{\lambda }g(z):=g ( z ) \ast \frac{z}{ ( z-1 ) ^{\lambda }}= \frac{z ( z^{\lambda -1}g(z) ) ^{(\lambda )}}{\lambda !}\quad (\lambda \in \mathbf{\mathbb{N}}_{0}, z\in { \mathbb{U}}). $$

Now, we define the Ruscheweyh derivative $\mathcal{D}^{\lambda }$ on the class of meromorphic harmonic functions. Let $\mathcal{D}_{\mathcal{H}}^{\lambda }:\mathcal{M}_{\mathcal{H}} ( k ) \rightarrow \mathcal{M}_{\mathcal{H}} ( k )$ denote the operator defined for a function $f=h+\overline{g}\in \mathcal{M}_{\mathcal{H}} ( k )$ by

$$\begin{aligned} \mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) :=& \frac{1}{z}+ ( -1 ) ^{\lambda }\mathcal{D}^{\lambda } \biggl( h ( z ) -\frac{1}{z} \biggr) +\overline{\mathcal{D}^{\lambda }g ( z ) } \\ =&\frac{1}{z}+ \biggl( f ( z ) -\frac{1}{z} \biggr) \ast \biggl( \frac{z}{ ( z-1 ) ^{\lambda }}+ \frac{\overline{z}}{ ( 1-\overline{z} ) ^{\lambda }} \biggr) \\ =&\frac{1}{z}+ ( -1 ) ^{\lambda }\sum_{n=k}^{ \infty } \lambda _{n}a_{n}z^{n}+\sum _{n=k}^{\infty }\lambda _{n} \overline{b_{n}} \overline{z}^{n}\quad ( z\in \mathbb{U} ) , \end{aligned}$$

where

$$ \lambda _{1}=1,\qquad \lambda _{n}:= \frac{ ( \lambda +1 ) \cdot \ldots \cdot (\lambda +n-1)}{(n-1)!} \quad ( n=2,3, \ldots ) . $$

(3)

It is clear that $\mathcal{D}_{\mathcal{H}}^{0}f=f$ and $\mathcal{D}_{\mathcal{H}}^{1}f=\mathcal{D}_{\mathcal{H}}f$.

Due to Janowski [13] (see also [9]) we define the class $\mathcal{M}_{\mathcal{H}}^{\lambda } ( k;M,N )$ of functions $f\in \mathcal{M}_{\mathcal{H}} ( k )$ that satisfy the following condition

$$ \frac{\mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) }{\mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) }\preceq \frac{1+Mz}{1+Nz},\quad {-N\leq M< N\leq 1.} $$

(4)

By $\mathcal{W}_{\mathcal{H}}^{\lambda } ( k;M,N )$ we denote the class of functions $f\in \mathcal{M}_{\mathcal{H}} ( k )$ such that

$$ z\mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) \preceq \frac{1+Mz}{1+Nz},\quad {-N \leq M< N\leq 1.} $$

Moreover, let us denote

$$\begin{aligned}& \mathcal{M}_{\mathcal{H}}^{\ast } ( k;M,N ) : = \mathcal{M}_{\mathcal{H}}^{0} ( k;M,N ) ,\qquad \mathcal{M}_{\mathcal{H}}^{c} ( k;M,N ) := \mathcal{M}_{\mathcal{H}}^{1} ( k;M,N ) , \\& \mathcal{M}_{\mathcal{H}}^{\ast }(\alpha ) : =\mathcal{M}_{ \mathcal{H}}^{\ast }(1,2 \alpha -1,1),\qquad \mathcal{M}_{\mathcal{H}}^{c}(\alpha ):= \mathcal{M}_{\mathcal{H}}^{c}(1,2\alpha -1,1). \end{aligned}$$

The classes $\mathcal{M}_{\mathcal{H}}^{\ast }:=\mathcal{M}_{\mathcal{H}}^{\ast }(0)$ and $\mathcal{M}_{\mathcal{H}}^{c}:=\mathcal{M}_{\mathcal{H}}^{c}(0)$ were studied in [3] (see also [9]). We see that the function $f\in \mathcal{M}_{\mathcal{H}}^{\ast }$ is starlike in $\mathbb{U} ( r )$ for all $r\in ( 0,1 \rangle$.

In this paper, we obtain some necessary and sufficient conditions for the defined classes of functions. In addition to finding certain analytic criteria, we obtain radii of starlikeness and convexity, and some topological properties for the defined classes of functions. Some applications of these results are also given.

2 Analytic criteria

To obtain the main results we need the following lemma.

Lemma 1

[8] A complex-valued function φ in ${\mathbb{U}}$ is weakly subordinate to a complex-valued univalent function Φ in ${\mathbb{U}}$ if and only if there exists a complex-valued function ω that maps ${\mathbb{U}}$ into oneself such that $\varphi (z)=\Phi (\omega (z))$, $z\in {\mathbb{U}}$.

Theorem 1

Let $f\in \mathcal{M}$ be of the form (1) and

$$ c_{n}=\lambda _{n} \bigl\{ n ( 1+N ) + ( 1+M ) \bigr\} ,\qquad d_{n}=\lambda _{n} \bigl\{ n ( 1+N ) - ( 1+M ) \bigr\} . $$

(5)

Then, $f\in \mathcal{M}_{\mathcal{H}}^{\lambda } ( k;M,N )$ if the condition

$$ \sum_{n=k}^{\infty } \bigl( c_{n} \vert a_{n} \vert +d_{n} \vert b_{n} \vert \bigr) \leq N-M $$

(6)

holds true.

Proof

It is easy to verify that

$$ \frac{c_{n}}{N-M}\geq n,\qquad \frac{d_{n}}{N-M}\geq n\quad ( n\in \mathbf{\mathbb{N}}_{k} ) . $$

Thus, by (6) we have

$$ \sum_{n=k}^{\infty } \bigl( n \vert a_{n} \vert +n \vert b_{n} \vert \bigr) \leq 1. $$

(7)

It is well known that the Jacobian of f is given by

$$ J_{f} ( z ) = \bigl\vert h^{\prime } ( z ) \bigr\vert ^{2}- \bigl\vert g^{\prime } ( z ) \bigr\vert ^{2} \quad ( z\in {\mathbb{U}} ) . $$

A function f is locally univalent and sense-preserving if the Jacobian of f is positive in ${\mathbb{U}}$. Lewy [15] proved that the converse is true for harmonic mappings. Since

$$\begin{aligned} \bigl\vert z^{2}J_{f} ( z ) \bigr\vert & = \bigl\vert z^{2}h^{ \prime } ( z ) \bigr\vert - \bigl\vert z^{2}g^{\prime } ( z ) \bigr\vert \\ &\geq 1-\sum _{n=k}^{\infty }n \vert a_{n} \vert \vert z \vert ^{n+2}-\sum_{n=k}^{\infty }n \vert b_{n} \vert \vert z \vert ^{n+2} \\ & \geq 1- \vert z \vert \sum_{n=k}^{\infty } \bigl( n \vert a_{n} \vert +n \vert b_{n} \vert \bigr) \geq 1- \vert z \vert >0 \quad ( z\in { \mathbb{U}} ) , \end{aligned}$$

we have that f is locally univalent and sense-preserving in ${ \mathbb{U}}$. To obtain univalence we assume that $w_{1},w_{2}\in {\mathbb{U}}$, $w_{1}\neq w_{2}$. Then,

$$ \biggl\vert \frac{w_{1}^{n}-w_{2}^{n}}{w_{1}-w_{2}} \biggr\vert = \Biggl\vert \sum _{l=1}^{n}w_{1}^{l-1}w_{2}^{n-l} \Biggr\vert \leq \sum_{l=1}^{n} \vert w_{1} \vert ^{l-1} \vert w_{2} \vert ^{n-l}\leq n\quad ( n\in { \mathbb{N}} ) $$

and by (7) we obtain

$$\begin{aligned} \bigl\vert f ( w_{1} ) -f ( w_{2} ) \bigr\vert & \geq \bigl\vert h ( w_{1} ) -h ( w_{2} ) \bigr\vert - \bigl\vert g ( w_{1} ) -g ( w_{2} ) \bigr\vert \\ & = \Biggl\vert \frac{1}{w_{1}}-\frac{1}{w_{2}}-\sum _{n=k}^{ \infty }a_{n} \bigl( w_{1}^{n}-w_{2}^{n} \bigr) \Biggr\vert - \Biggl\vert \sum_{n=k}^{\infty } \overline{b_{n} \bigl( w_{1}^{n}-w_{2}^{n} \bigr) } \Biggr\vert \\ & \geq \frac{ \vert w_{1}-w_{2} \vert }{ \vert w_{1}w_{2} \vert }- \sum_{n=k}^{\infty } \vert a_{n} \vert \bigl\vert w_{1}^{n}-w_{2}^{n} \bigr\vert -\sum_{n=k}^{\infty } \vert b_{n} \vert \bigl\vert w_{1}^{n}-w_{2}^{n} \bigr\vert \\ & = \vert w_{1}-w_{2} \vert \Biggl( \frac{1}{ \vert w_{1}w_{2} \vert }- \sum_{n=k}^{ \infty } \vert a_{n} \vert \biggl\vert \frac{w_{1}^{n}-w_{2}^{n}}{w_{1}-w_{2}} \biggr\vert -\sum _{n=k}^{\infty } \vert b_{n} \vert \biggl\vert \frac{w_{1}^{n}-w_{2}^{n}}{w_{1}-w_{2}} \biggr\vert \Biggr) \\ & > \vert w_{1}-w_{2} \vert \Biggl( 1-\sum _{n=k}^{ \infty }n \vert a_{n} \vert -\sum _{n=k}^{\infty }n \vert b_{n} \vert \Biggr) \geq 0. \end{aligned}$$

Thus, $f\in \mathcal{M}_{\mathcal{H}} ( k )$ and by Lemma 1 we obtain that $f\in \mathcal{M}_{\mathcal{H}}^{\ast } ( k;M,N )$ if and only if there exists a complex-valued function χ bounded by 1 in ${\mathbb{U}}$ for which

$$ \frac{\mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) }{\mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) }= \frac{1+M\chi (z)}{1+N\chi (z)}\quad ( z\in {\mathbb{U}} ) , $$

or equivalently

$$ \biggl\vert \frac{\mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) -\mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) }{N\mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) -M\mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) ( z ) } \biggr\vert < 1\quad ( z\in {\mathbb{U}} ) . $$

(8)

Therefore, we need to show that

$$ \bigl\vert \mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) - \mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) \bigr\vert - \bigl\vert N\mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) - \mathcal{D}_{ \mathcal{H}}^{\lambda }f ( z ) \bigr\vert < 0\quad ( z\in { \mathbb{U}} ) . $$

Putting $\vert z \vert =r$ ($0< r<1$) we obtain

$$\begin{aligned} & \bigl\vert \mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) -\mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) \bigr\vert - \bigl\vert N \mathcal{D}_{\mathcal{H}}^{\lambda +1}f ( z ) -\mathcal{D}_{\mathcal{H}}^{\lambda }f ( z ) \bigr\vert \\ &\quad = \Biggl\vert \sum_{n=k}^{\infty } ( -1 ) ^{ \lambda }\lambda _{n} ( n+1 ) a_{n}z^{n}-\sum _{n=k}^{ \infty }\lambda _{n} ( n-1 ) \overline{b_{n}}\overline{z}^{n} \Biggr\vert \\ & \qquad {}- \Biggl\vert ( N-M ) \frac{1}{z}-\sum_{n=k}^{ \infty } ( -1 ) ^{\lambda }\lambda _{n} ( Nn+M ) a_{n}z^{n}+ \sum_{n=k}^{\infty }\lambda _{n} ( Nn-M ) \overline{b_{n}}\overline{z}^{n} \Biggr\vert \\ & \quad \leq \sum_{n=k}^{\infty }\lambda _{n} ( n+1 ) \vert a_{n} \vert r^{n}+\sum _{n=k}^{\infty } \lambda _{n} ( n-1 ) \vert b_{n} \vert r^{n}- ( N-M ) \frac{1}{r} \\ & \qquad {}+\sum_{n=k}^{\infty }\lambda _{n} ( Nn+M ) \vert a_{n} \vert r^{n}+\sum _{n=k}^{\infty } \lambda _{n} ( Nn-M ) \vert b_{n} \vert r^{n} \\ & \quad \leq \frac{1}{r} \Biggl\{ \sum_{n=k}^{\infty } \bigl( c_{n} \vert a_{n} \vert +d_{n} \vert b_{n} \vert \bigr) r^{n+1}- ( N-M ) \Biggr\} < 0, \end{aligned}$$

which implies $f\in \mathcal{M}_{\mathcal{H}}^{\lambda } ( k;M,N )$. □

Let $\mathcal{T}_{\eta }^{\lambda } ( k )$ be the class of functions $f=h+\overline{g}\in \mathcal{M} ( k )$ with varying coefficients (e.g., see [12]) so that

$$ f(z)=h(z)+\overline{g(z)}=\frac{1}{z}+ ( -1 ) ^{\lambda } \sum _{n=k}^{\infty }e^{-i ( n+1 ) \eta } \vert a_{n} \vert z^{n}-\sum_{n=k}^{\infty }e^{i ( n-1 ) \eta } \vert b_{n} \vert \overline{{z}}^{n} $$

(9)

and let

$$\begin{aligned}& \mathcal{M}_{\eta }^{\lambda } ( k;M,N ) : =\mathcal{T}_{ \eta }^{\lambda } ( k ) \cap \mathcal{M}_{\mathcal{H}}^{ \lambda } ( k;M,N ) ,\qquad \mathcal{W}_{\mathcal{\eta }}^{ \lambda } ( k;M,N ) :=\mathcal{T}_{\eta }^{\lambda } ( k ) \cap \mathcal{W}_{\mathcal{H}}^{\lambda } ( k;M,N ) , \\& \mathcal{M}_{\mathcal{\eta }}^{\ast } ( k;M,N ) : = \mathcal{T}_{\eta }^{0} ( k ) \cap \mathcal{M}_{\mathcal{H}}^{\ast } ( k;M,N ) ,\qquad \mathcal{M}_{\mathcal{\eta }}^{c} ( k;M,N ) :=\mathcal{T}_{\eta }^{1} ( k ) \cap \mathcal{M}_{ \mathcal{H}}^{c} ( k;M,N ) . \end{aligned}$$

The sufficient coefficient bound given in Theorem 1 is also necessary for functions to be in the class $\mathcal{M}_{\eta }^{\lambda } ( k;M,N )$, as stated in the following theorem.

Theorem 2

Let $f\in \mathcal{T}_{\eta }^{\lambda }$ be a function of the form (1). Then, $f\in \mathcal{M}_{\eta }^{\lambda } ( k;M,N )$ if and only if the condition (6) holds true.

Proof

By Theorem 1 we need to prove the “only if” part. Let $f\in \mathcal{M}_{\eta }^{\ast } ( k;M,N )$. Then, by (8) we obtain

$$ \biggl\vert \frac{\sum_{n=k}^{\infty }\lambda _{n} \{ ( -1 ) ^{\lambda } ( n+1 ) a_{n}z^{n+1}- ( n-1 ) \overline{b_{n}}z\overline{z}^{n} \} }{ ( N-M ) -\sum_{n=k}^{\infty }\lambda _{n} \{ ( -1 ) ^{\lambda } ( Nn+M ) a_{n}z^{n+1}- ( Nn-M ) \overline{b_{n}}\overline{zz}^{n} \} } \biggr\vert < 1 \quad (z\in {\mathbb{U}}). $$

Thus, by (9) for $z=re^{i\eta }$ ($0< r<1$), we have

$$ \frac{\sum_{n=k}^{\infty }\lambda _{n} \{ ( n+1 ) \vert a_{n} \vert + ( n-1 ) \vert b_{n} \vert \} r^{n+1}}{ ( N-M ) -\sum_{n=k}^{\infty }\lambda _{n} \{ ( Nn+M ) \vert a_{n} \vert + ( Nn-M ) \vert b_{n} \vert \} r^{n+1}}< 1. $$

(10)

The denominator of the left-hand side cannot vanish for $r\in ( 0,1 )$. Also, it is positive for $r=0$, and in consequence for $r\in ( 0,1 )$. Thus, by (10) we have

$$ \sum_{n=k}^{\infty } \bigl( c_{n} \vert a_{n} \vert +d_{n} \vert b_{n} \vert \bigr) r^{n+1}< N-M \quad (0< r< 1). $$

(11)

The sequence of partial sums $\{ S_{n} \}$ related to the series $\sum_{n=k}^{\infty } ( \alpha _{n} \vert a_{n} \vert +\beta _{n} \vert b_{n} \vert )$ is a nondecreasing sequence. Moreover, by (11) it is bounded by $N-M$. Hence, the sequence $\{ S_{n} \}$ is convergent and $\sum_{n=k}^{\infty } ( \alpha _{n} \vert a_{n} \vert +\beta _{n} \vert b_{n} \vert ) =\lim_{n \rightarrow \infty }S_{n}\leq N-M$, which gives (6). □

Analogously as Theorem 2 we can prove the following theorem.

Theorem 3

Let $f\in \mathcal{T}_{\eta }^{\lambda } ( k )$ be a function of the form (1). Then, $f\in \mathcal{W}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$ if and only if

$$ \sum_{n=k}^{\infty }\lambda _{n} \bigl( \vert a_{n} \vert + \vert b_{n} \vert \bigr) \leq \frac{N-M}{1+N}. $$

(12)

By Theorems 2 and 3 we have the following corollary.

Corollary 1

Let $a=\frac{1+M}{1+N}$ and

$$\begin{aligned}& \phi ( z ) = \frac{1}{z}+\sum_{n=k}^{ \infty } \biggl( \frac{1}{n+a}z^{n}+\frac{1}{n-a}\overline{z}^{n} \biggr) \quad ( z\in {\mathbb{U}} ) , \\& \omega ( z ) = \frac{1}{z}+\sum_{n=k}^{ \infty } \bigl( ( n+a ) z^{n}+ ( n-a ) \overline{z}^{n} \bigr)\quad ( z\in {\mathbb{U}} ) . \end{aligned}$$

Then,

$$\begin{aligned}& f \in \mathcal{W}_{\mathcal{\eta }}^{\lambda } ( k;M,N ) \quad \Leftrightarrow\quad f\ast \phi \in \mathcal{M}_{\mathcal{\eta }}^{ \lambda } ( k;M,N ) , \\& f \in \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N ) \quad \Leftrightarrow\quad f\ast \omega \in \mathcal{W}_{\mathcal{\eta }}^{ \lambda } ( k;M,N ) . \end{aligned}$$

In particular,

$$ \mathcal{W}_{\mathcal{\eta }}^{\lambda +1}(-1,N)=\mathcal{M}_{ \mathcal{\eta }}^{\lambda } ( -1,N ) . $$

Remark 1

If we put $n=0$ or $n=1$ in Theorems 1 and 2, then we obtain similar results for the classes $\mathcal{M}_{\mathcal{\eta }}^{\ast } ( k;M,N )$ and $\mathcal{M}_{\mathcal{\eta }}^{c} ( k;M,N )$.

3 Radii of convexity and starlikeness of order α

By using condition (2) we generalize the definition of starlikeness of meromorphic harmonic functions. We say that a function $f\in \mathcal{T}_{\eta }^{\lambda } ( k )$ is starlike of order α in ${\mathbb{U}}(r)$ if

$$ \frac{\partial }{\partial t} \bigl( \arg f \bigl( \rho e^{it} \bigr) \bigr) < \alpha ,\quad 0< \rho < r< 1, 0\leq t\leq 2\pi . $$

(13)

Also, a function $f\in \mathcal{T}_{\eta }^{\lambda } ( k )$ is said to be convex of order α in ${\mathbb{U}}(r)$ if

$$ \frac{\partial }{\partial t} \biggl( \arg \frac{\partial }{\partial t}f \bigl( \rho e^{it} \bigr) \biggr) < \alpha , \quad 0< \rho < r< 1, 0 \leq t\leq 2\pi . $$

It is easy to verify that for a function $f\in \mathcal{T}_{\eta }^{\lambda } ( k )$ the condition (13) is equivalent to the following

$$ \mathrm{Re} {\frac{\mathcal{D}_{\mathcal{H}}{f}(z)}{f(z)}}>\alpha \quad \bigl(z \in {\mathbb{U}}(r) \bigr) $$

or equivalently

$$ \biggl\vert { \frac{\mathcal{D}_{\mathcal{H}}{f}(z)-f(z)}{\mathcal{D}_{\mathcal{H}}{f}(z)- ( 2\alpha -1 ) f(z)}} \biggr\vert < 1 \quad \bigl(z \in {\mathbb{U}}(r)\bigr). $$

(14)

Let $\mathcal{B}$ be a subclass of the class $\mathcal{T}_{\eta }^{\lambda } ( k )$. We define the radius of starlikeness $R_{\alpha }^{\ast }(\mathcal{B})$ and the radius of convexity $R_{\alpha }^{c}(\mathcal{B})$ for the class $\mathcal{B}$ by

$$\begin{aligned} R_{\alpha }^{\ast }(\mathcal{B})& :=\inf_{f\in \mathcal{B}} ( \sup \bigl\{ r\in \bigl(0,1]:f \text{ is starlike of order }\alpha\text{ in } {\mathbb{U}}(r) \bigr\} \bigr) , \\ R_{\alpha }^{c}(\mathcal{B})& :=\inf_{f\in \mathcal{B}} ( \sup \bigl\{ r\in \bigl(0,1]:f \text{ is convex of order }\alpha \text{ in }{\mathbb{U}}(r) \bigr\} \bigr) . \end{aligned}$$

Theorem 4

$$ {R_{\alpha }^{\ast }\bigl(\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N ) \bigr)}=\inf_{n\in {\mathbb{N}}_{k}} \biggl( \frac{1-\alpha }{N-M}\min \biggl\{ \frac{c_{n}}{n+\alpha },\frac{d_{n}}{n-\alpha } \biggr\} \biggr) ^{{\frac{1}{n+1}}}, $$

(15)

where $c_{n}$ and $d_{n}$ are defined by (5).

Proof

Let $f\in \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( M,N )$ be of the form (1). Then, putting $\vert z \vert =r<1$ we have

$$\begin{aligned} \biggl\vert \frac{\mathcal{D}_{\mathcal{H}}{f}(z)-f(z)}{\mathcal{D}_{\mathcal{H}}{f}(z)- ( 2\alpha -1 ) f(z)} \biggr\vert & = \biggl\vert { \frac{\sum_{n=k}^{\infty }(n+1)a_{n}z^{n}-\sum_{n=k}^{\infty }(n-1)\overline{b_{n}\overline{z}^{n}}}{\frac{2 ( 1-\alpha ) }{z}-\sum_{n=k}^{\infty }(n+2\alpha -1)a_{n}z^{n}+\sum_{n=k}^{\infty }(n-2\alpha +1)\overline{b_{n}\overline{z}^{n}}}} \biggr\vert \\ & \leq { \frac{\sum_{n=k}^{\infty } ( (n+1) \vert a_{n} \vert +(n-1) \vert b_{n} \vert ) r^{n+1}}{2 ( 1-\alpha ) -\sum_{n=k}^{\infty } ( (n+2\alpha -1) \vert a_{n} \vert +(n-2\alpha +1) \vert b_{n} \vert ) r^{n+1}}.} \end{aligned}$$

Thus, the condition (14) is true if

$$ \sum_{n=k}^{\infty } \biggl( \frac{n+\alpha }{1-\alpha } \vert a_{n} \vert +\frac{n-\alpha }{1-\alpha } \vert b_{n} \vert \biggr) r^{n+1}\leq 1. $$

(16)

By Theorem 2, we have

$$ \sum_{n=k}^{\infty } \biggl( \frac{c_{n}}{N-M} \vert a_{n} \vert +\frac{d_{n}}{N-M} \vert b_{n} \vert \biggr) \leq 1, $$

(17)

where $c_{n}$ and $d_{n}$ are defined by (5). Thus, the condition (16) is true if

$$ \frac{n+\alpha }{1-\alpha }r^{n+1}\leq \frac{c_{n}}{N-M},\qquad \frac{n-\alpha }{1-\alpha }r^{n+1}\leq \frac{d_{n}}{N-M}\quad (n\in { \mathbb{N}}_{k}), $$

that is, if

$$ r\leq \biggl( \frac{1-\alpha }{N-M}\min \biggl\{ \frac{c_{n}}{n+\alpha }, \frac{d_{n}}{n-\alpha } \biggr\} \biggr) ^{{\frac{1}{n+1}}} \quad (n \in {\mathbb{N}}_{k}). $$

It follows that the function f is starlike of order α in the disk ${\mathbb{U}}(r^{\ast })$, where

$$ r^{\ast }:{{=\inf_{n\in {\mathbb{N}}_{k}} \biggl( \frac{1-\alpha }{N-M}\min \biggl\{ \frac{c_{n}}{n+\alpha },\frac{d_{n}}{n-\alpha } \biggr\} \biggr) ^{{\frac{1}{n+1}}}.}} $$

(18)

The radii of starlikeness $r^{\ast } ( h_{n} )$, $r^{\ast } ( g_{n} )$ of functions $h_{n}$, $g_{n}$ ($n\in \mathbb{N}$) of the form

$$ h_{n}(z)=\frac{1}{z}+ \frac{ ( -1 ) ^{\lambda } ( N-M ) }{c_{n}e^{i ( n+1 ) \eta }}z^{n},\qquad g_{n}(z)=\frac{1}{z}- \frac{N-M}{d_{n}e^{i ( 1-n ) \eta }}\overline{z}^{n} \quad (n\in { \mathbb{N}}_{k}, z\in {\mathbb{U}}) $$

are given by

$$ r^{\ast } ( h_{n} ) = \biggl( \frac{1-\alpha }{n+\alpha } \frac{c_{n}}{N-M} \biggr) ^{\frac{1}{n+1}}, \qquad r^{\ast } ( g_{n} ) ={{ \biggl( \frac{1-\alpha }{n-\alpha }\frac{d_{n}}{N-M} \biggr) ^{ \frac{1}{n+1}}.}} $$

Therefore, the radius $r^{\ast }$ given by (18) cannot be larger. Thus, we have (15). □

The following result may be proved in much the same way as Theorem 4.

Theorem 5

Let $c_{n}$ and $d_{n}$ be defined by (5). Then,

$$ {R_{\alpha }^{c}\bigl(\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N ) \bigr)}=\inf_{n\in {\mathbb{N}}_{k}} \biggl( \frac{1-\alpha }{n ( N-M ) }\min \biggl\{ \frac{c_{n}}{n+\alpha }, \frac{d_{n}}{n-\alpha } \biggr\} \biggr) ^{{\frac{1}{n+1}}}. $$

If we put $n=0$ or $n=1$ in Theorems 4 and 5 we obtain the following results.

Corollary 2

$$\begin{aligned}& {R_{\alpha }^{\ast }\bigl(\mathcal{M}_{\mathcal{\eta }}^{\ast } ( k;M,N ) \bigr)} \\& \quad = {R_{\alpha }^{c}\bigl(\mathcal{M}_{\mathcal{\eta }}^{c} ( k;M,N ) \bigr)} \\& \quad = \inf_{n\in {\mathbb{N}}_{k}} \biggl( \frac{1-\alpha }{N-M} \min \biggl\{ \frac{n ( 1+N ) + ( 1+M ) }{n+\alpha }, \frac{n ( 1+N ) - ( 1+M ) }{n-\alpha } \biggr\} \biggr) ^{{\frac{1}{n+1}}}, \\& {R_{\alpha }^{c}\bigl(\mathcal{M}_{\mathcal{\eta }}^{\ast } ( k;M,N ) \bigr)} \\& \quad = \inf_{n\in {\mathbb{N}}_{k}} \biggl( \frac{1-\alpha }{n ( N-M ) }\min \biggl\{ \frac{n ( 1+N ) + ( 1+M ) }{n+\alpha }, \frac{n ( 1+N ) - ( 1+M ) }{n-\alpha } \biggr\} \biggr) ^{{\frac{1}{n+1}}}, \\& {R_{\alpha }^{\ast }\bigl(\mathcal{M}_{\mathcal{\eta }}^{c} ( k;M,N ) \bigr)} \\& \quad = \inf_{n\in {\mathbb{N}}_{k}} \biggl( \frac{n ( 1-\alpha ) }{N-M}\min \biggl\{ \frac{n ( 1+N ) + ( 1+M ) }{n+\alpha },\frac{n ( 1+N ) - ( 1+M ) }{n-\alpha } \biggr\} \biggr) ^{{\frac{1}{n+1}}}. \end{aligned}$$

4 Topological properties

Let us consider the usual topology on $\mathcal{ M}_{\mathcal{H}} ( k )$ defined by a metric in which a sequence $\{ f_{n} \}$ in $\mathcal{ M}_{\mathcal{H}} ( k )$ converges to f if and only if it converges to f uniformly on each compact subset of ${\mathbb{U}}$. It follows from the theorems of Weierstrass and Montel that this topological space is complete.

Let $\mathcal{B}$ be a subclass of the class $\mathcal{M}_{\mathcal{H}} ( k )$. We say that a function $f\in \mathcal{B}$ is an extreme point of $\mathcal{B}$ if it cannot be represented as a nondegenerate, convex, and linear combination of two function from $\mathcal{B}$. We denote by $E\mathcal{B}$ the set of all extreme points of $\mathcal{B}$. We have that $E\mathcal{B}\subset \mathcal{B}$.

A class $\mathcal{B}$ is called convex if any convex linear combination of two functions from $\mathcal{B}$ belongs to $\mathcal{B}$. We denote by $\overline{co}\mathcal{B}$ the closed convex hull of $\mathcal{B}$, i.e., the intersection of all closed, convex subsets of ${\mathcal{M}}$ that contain $\mathcal{B}$..

A real-valued functional $\mathcal{D}:\mathcal{M}_{\mathcal{H}} ( k ) \rightarrow \mathbb{R}$ is called convex on a convex class $\mathcal{B}\subset \mathcal{M}_{\mathcal{H}} ( k )$ if for $f,g\in \mathcal{B}$ and $0\leq \lambda \leq 1$ we have

$$ \mathcal{D} \bigl( \gamma f+ ( 1-\gamma ) g \bigr) \leq \gamma \mathcal{D} ( f ) + ( 1-\gamma ) \mathcal{D} ( g ) . $$

From the Krein–Milman theorem (see [14]) we have the following lemma.

Lemma 2

Let $\mathcal{B}$ be a nonempty, compact, and convex subclass of the class $\mathcal{M}_{\mathcal{H}} ( k )$ and $\mathcal{D}:\mathcal{M}_{\mathcal{H}} ( k ) \rightarrow \mathbb{R}$ be a real-valued, continuous, and convex functional on $\mathcal{B}$. Then,

$$ \mathcal{B}=\overline{co}E\mathcal{B} $$

and

$$ \max \bigl\{ \mathcal{D}(f):f\in \mathcal{B} \bigr\} =\max \bigl\{ \mathcal{D}(f):f\in E\mathcal{B} \bigr\} . $$

Moreover, from Montel’s theorem we obtain the following lemma.

Lemma 3

A class $\mathcal{B}\subset \mathcal{M}_{\mathcal{H}} ( k )$ is compact if and only if $\mathcal{B}$ is closed and locally uniformly bounded.

Theorem 6

The class $\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$ is a compact and convex subclass of $\mathcal{M}_{\mathcal{H}} ( k )$.

Proof

Let ${0\leq \lambda \leq 1}$ and $f_{1},f_{2}\in \mathcal{M}_{ \mathcal{\eta }}^{\lambda } ( k;M,N )$ be functions of the form

$$ f_{l}(z)=\frac{1}{z}+\sum_{n=k}^{\infty } \bigl( a_{l,n}z^{n}+\overline{b_{l,n}} \overline{z}^{n} \bigr)\quad ( z\in { \mathbb{U}},l\in { \mathbb{N}} ) . $$

(19)

Then, we have

$$\begin{aligned}& \lambda f_{1}(z)+(1-\lambda )f_{2} ( z ) \\& \quad = \frac{1}{z}+\sum_{n=k}^{\infty } \bigl\{ \bigl( \lambda a_{1,n}+ ( 1-\lambda ) a_{2,n} \bigr) z^{n}+ \overline{ \bigl( \lambda b_{1,n}+ ( 1-\lambda ) b_{2,n} \bigr) z^{n}} \bigr\} . \end{aligned}$$

Moreover, by Theorem 2 we obtain

$$\begin{aligned}& \sum_{n=k}^{\infty } \bigl\{ c_{n} \bigl\vert \gamma a_{1,n}+ ( 1-\gamma ) a_{2,n} \bigr\vert +d_{n} \bigl\vert \gamma b_{1,n}+ ( 1-\gamma ) b_{2,n} \bigr\vert \bigr\} \\& \quad \leq \gamma \sum_{n=k}^{\infty } \bigl\{ c_{n} \vert a_{1,n} \vert +d_{n} \vert b_{1,n} \vert \bigr\} + ( 1- \gamma ) \sum _{n=k}^{\infty } \bigl\{ c_{n} \vert a_{2,n} \vert +d_{n} \vert b_{2,n} \vert \bigr\} \\& \quad \leq \gamma ( N-M ) + ( 1-\gamma ) ( N-M ) =N-M. \end{aligned}$$

Thus, the function ${\varphi }=\lambda f_{1}+(1-\lambda )f_{2}$ belongs to the class $\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$ and, in consequence, the class is convex.

The class is locally uniformly bounded if for each r, R, $0< r< R<1$, there is a real constant $L=L ( r,R )$ so that

$$ \bigl\vert f(z) \bigr\vert \leq L \quad \bigl( f\in \mathcal{F}, r\leq \vert z \vert \leq R \bigr) . $$

Let $f\in \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$, $0< r\leq \vert z \vert \leq R<1$. Then, by Theorem 2, we have

$$ \bigl\vert f(z) \bigr\vert \leq \frac{1}{r}+\sum _{n=k}^{ \infty } \bigl( \vert a_{n} \vert + \vert b_{n} \vert \bigr) R^{n}\leq \frac{1}{r}+\sum _{n=k}^{ \infty } \bigl( c_{n} \vert a_{n} \vert +d_{n} \vert b_{n} \vert \bigr) \leq \frac{1}{r}+ ( N-M ) =:L. $$

This implies that the class $\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$ is locally uniformly bounded. Next, we show that it is closed. Let $f_{l}$ and f be given by (19) and (1), respectively. By Theorem 2 we obtain

$$ \sum_{n=k}^{\infty } \bigl( c_{n} \vert a_{l,n} \vert +d_{n} \vert b_{l,n} \vert \bigr) \leq N-M\quad ( l\in {\mathbb{N}} ) . $$

(20)

If $f_{l}\rightarrow f$, then we obtain that $a_{l,n}\rightarrow a_{n}$ and $b_{l,n}\rightarrow b_{n}$ as $l\rightarrow \infty$ ($n\in {\mathbb{N}}_{k}$). The sequence of partial sums $\{ S_{n} \}$ associated with the series $\sum_{n=k}^{\infty } ( c_{n} \vert a_{n} \vert +d_{n} \vert b_{n} \vert )$ is a nondecreasing sequence. Moreover, by (20) it is bounded by $N-M$. Therefore, the sequence $\{ S_{n} \}$ is convergent and $\sum_{n=k}^{\infty } ( c_{n} \vert a_{n} \vert +d_{n} \vert b_{n} \vert ) =\lim_{n \rightarrow \infty }S_{n}\leq N-M$. This gives the condition (6), and, in consequence, $f\in \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$, which completes the proof. □

Theorem 7

$$ E\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N ) = \{ h_{n}: n\in \mathbb{N}_{k-1} \} \cup \{ g_{n}: n\in {\mathbb{N}}_{k} \} , $$

where $h_{k-1}(z)=\frac{1}{z}$ and

$$ h_{n}(z)=\frac{1}{z}+ \frac{ ( -1 ) ^{\lambda } ( N-M ) }{c_{n}e^{i ( n+1 ) \eta }}z^{n},\qquad g_{n}(z)=\frac{1}{z}- \frac{N-M}{d_{n}e^{i ( 1-n ) \eta }}\overline{z}^{n} \quad (n\in { \mathbb{N}}_{k}, z\in {\mathbb{U}}). $$

(21)

Proof

Let $f_{1},f_{2}\in \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$ be functions of the form (6), $g_{n}=\lambda f_{1}+ ( 1-\lambda ) f_{2}$ with $0<\lambda <1$. Then, by (6) we obtain $\vert b_{1,n} \vert = \vert b_{2,n} \vert =\frac{N-M}{\beta _{n}}$. Thus, $a_{1,l}=a_{2,l}=0$ for $l\in {\mathbb{N}}_{k}$ and $b_{1,l}=b_{2,l}=0$ for $l\in {\mathbb{N}}_{k}\diagdown \{ n \}$. This means that $g_{n}=f_{1}=f_{2}$, and, in consequence, $g_{n}\in E\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$. In the same way, we show that the functions $h_{n}$ of the form (21) are the extreme points of the class $\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$. Now, let $f\in E\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$ be not of the form (21). Then, there exists $r\in {\mathbb{N}}_{k}$ such that

$$ 0< \vert a_{r} \vert < \frac{N-M}{\alpha _{r}}\quad \text{or}\quad 0< \vert b_{r} \vert < \frac{N-M}{\beta _{r}}. $$

If $0< \vert a_{r} \vert <\frac{N-M}{\alpha _{r}}$, then for

$$ \lambda =\frac{\alpha _{r} \vert a_{r} \vert }{N-M},\qquad \varphi =\frac{1}{1-\lambda } ( f-\lambda h_{r} ) , $$

we have that $0<\lambda <1$, $h_{r}\neq \varphi$ and $f=\lambda h_{r}+ ( 1-\lambda ) \varphi$. Thus, $f\notin E\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$. Analogously, if $0< \vert b_{r} \vert <\frac{N-M}{\beta _{n}}$, then for

$$ \lambda =\frac{\beta _{r} \vert b_{r} \vert }{N-M},\qquad \phi = \frac{1}{1-\lambda } ( f-\lambda g_{r} ) , $$

we have that $0<\lambda <1$, $g_{r}\neq \phi$ and $f=\lambda g_{r}+ ( 1-\lambda ) \phi$. Thus, $f\notin E\mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$, and the proof is completed. □

5 Applications of extreme points

If the class $\mathcal{B}= \{ f_{n}\in \mathcal{M}_{\mathcal{H}} ( k ) : n\in \mathbf{\mathbb{N} } \}$ is locally uniformly bounded, then

$$ \overline{co}\mathcal{B}= \Biggl\{ \sum_{n=1}^{\infty } \lambda _{n}f_{n}: \sum_{n=1}^{\infty } \lambda _{n}=1, \lambda _{n}\geq 0 ( n\in \mathbf{\mathbb{N} } ) \Biggr\} . $$

Thus, by Lemma 2 and Theorem 7 we obtain

Corollary 3

$$ \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N ) = \Biggl\{ \sum _{n=k-1}^{\infty } ( \gamma _{n}h_{n}+ \delta _{n}g_{n} ) : \sum_{n=k-1}^{\infty } ( \gamma _{n}+\delta _{n} ) =1 ( \delta _{k-1}=0, \gamma _{n},\delta _{n}\geq 0 ) \Biggr\} , $$

where $h_{n}$, $g_{n}$ are defined by (21).

It is easy to show that the following real-valued functionals are convex and continuous on $\mathcal{M}_{\mathcal{H}} ( k )$:

$$\begin{aligned}& \mathcal{D} ( f ) =a_{n},\qquad \mathcal{D} ( f ) =b_{n},\qquad \mathcal{D} ( f ) = \bigl\vert f ( z ) \bigr\vert , \qquad \mathcal{D} ( f ) = \bigl\vert \mathcal{D}_{\mathcal{H}}^{ \lambda }f ( z ) \bigr\vert , \\& \mathcal{D} ( f ) = \biggl( \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert f \bigl( re^{i\theta } \bigr) \bigr\vert ^{\gamma }\,d\theta \biggr) ^{1/\gamma }\quad \bigl( f\in {\mathcal{M}_{\mathcal{H}} ( k ) } \bigr) . \end{aligned}$$

for each fixed value of $n\in {\mathbb{N}}_{k}$, $z\in {\mathbb{U}}$, $\gamma \geq 1$, $0< r<1$. Thus, by Lemma 2 and Theorem 7 we have the following results.

Corollary 4

Let $f\in \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$ be a function of the form (1), $0< r<1$, $\gamma \geq 1$. Then,

$$\begin{aligned} & \vert a_{n} \vert \leq {\frac{N-M}{c_{n}}, } \qquad \vert b_{n} \vert \leq {\frac{N-M}{d_{n}}}\quad (n\in { \mathbb{N}}_{k}), \\ & \frac{1}{r}-\frac{N-M}{d_{k}}r^{k}\leq \bigl\vert f(z) \bigr\vert \leq \frac{1}{r}+\frac{N-M}{d_{k}}r^{k}\quad \bigl( \vert z \vert =r \bigr) , \\ & \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert f\bigl(re^{i\theta } \bigr) \bigr\vert ^{\gamma }\,d\theta \leq \frac{1}{2\pi } \int _{0}^{2 \pi } \bigl\vert h_{1} \bigl(re^{i\theta }\bigr) \bigr\vert ^{\gamma }\,d\theta , \\ & \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert \mathcal{D}_{ \mathcal{H}}^{\lambda }f \bigl(re^{i\theta }\bigr) \bigr\vert ^{\gamma }\,d\theta \leq \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert \mathcal{D}_{ \mathcal{H}}^{\lambda }h_{1} \bigl(re^{i\theta }\bigr) \bigr\vert ^{\gamma }\,d\theta , \end{aligned}$$

where $c_{n}$, $d_{n}$ are defined by (5). The results are sharp with extremal functions $h_{n}$, $g_{n}$ of the form (21).

Corollary 5

If $f\in \mathcal{M}_{\mathcal{\eta }}^{\lambda } ( k;M,N )$, then

$$ {\mathbb{U}} ( r ) \subset f ( {\mathbb{U}} ) , $$

where

$$ r=1- \frac{N-M}{k^{\lambda +1} ( 1+N ) -k^{\lambda } ( 1+M ) }. $$

Remark 2

If we put $n=0$ or $n=1$ in Corrolaries 3 and 4 we obtain similar results for the classes $\mathcal{M}_{\mathcal{\eta }}^{\ast } ( k;M,N )$ and $\mathcal{M}_{\mathcal{\eta }}^{c} ( k;M,N )$.

By using Corollary 1 and the results above we obtain the corollaries listed below.

Corollary 6

The class $\mathcal{W}_{\mathcal{\eta }} ( k;M,N )$ is a convex and compact subset of $\mathcal{M}_{\mathcal{H}} ( k )$. Moreover,

$$ E\mathcal{W}_{\mathcal{\eta }} ( k;M,N ) = \{ h_{n}: n\in \mathbb{N}_{k-1} \} \cup \{ g_{n}: n\in { \mathbb{N}}_{k} \} $$

and

$$ \mathcal{W}_{\mathcal{\eta }} ( k;M,N ) = \Biggl\{ \sum _{n=1}^{ \infty } ( \gamma _{n}h_{n}+ \delta _{n}g_{n} ) : \sum_{n=1}^{ \infty } ( \gamma _{n}+\delta _{n} ) =1, \delta _{1}=0, \gamma _{n},\delta _{n}\geq 0 ( n\in \mathbb{N} ) \Biggr\} , $$

where $h_{k-1}(z)=z$ and

$$ h_{n}(z)=z+\frac{N-M}{ ( 1+N ) n}z^{n},\qquad g_{n}(z)=z- \frac{N-M}{ ( 1+N ) n}\overline{z}^{n} \quad (z\in {\mathbb{U}}). $$

(22)

Corollary 7

If $f\in \mathcal{W}_{\mathcal{\eta }} ( k;M,N )$ is of the form (1), then

$$\begin{aligned} & \vert a_{n} \vert \leq \frac{N-M}{ ( 1+N ) n}, \qquad \vert b_{n} \vert \leq \frac{N-M}{ ( 1+N ) n} \quad ( n\in \mathbb{N} ) , \\ & \frac{1}{r}-\frac{N-M}{ ( 1+N ) k}r^{k}\leq \bigl\vert f(z) \bigr\vert \leq \frac{1}{r}+\frac{N-M}{ ( 1+N ) k}r^{k}\quad \bigl( \vert z \vert =r< 1 \bigr) , \\ & \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert f\bigl(re^{i\theta } \bigr) \bigr\vert ^{\gamma }\,d\theta \leq \frac{1}{2\pi } \int _{0}^{2 \pi } \bigl\vert h_{2} \bigl(re^{i\theta }\bigr) \bigr\vert ^{\lambda }\,d\theta , \\ & \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert \mathcal{D}_{ \mathcal{H}}^{\lambda }f \bigl(re^{i\theta }\bigr) \bigr\vert ^{\gamma }\,d\theta \leq \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert \mathcal{D}_{ \mathcal{H}}^{\lambda }h_{2} \bigl(re^{i\theta }\bigr) \bigr\vert ^{\gamma }\,d\theta . \end{aligned}$$

The results are sharp with extremal functions $h_{n}$, $g_{n}$ of the form (22).

Corollary 8

If $f\in \mathcal{W}_{\mathcal{\eta }} ( k;M,N )$, then

$$ {\mathbb{U}} \biggl( 1-\frac{N-M}{ ( 1+N ) k} \biggr) \subset f ( {\mathbb{U}} ) . $$