Subclass of Harmonic Univalent Functions Associated With the Mittag-Leffler type functions

TL;DR

This paper introduces a new subclass of harmonic functions in the unit disc based on Mittag-Leffler type functions, analyzing their coefficient conditions, extreme points, distortion bounds, and convex combinations.

Contribution

The paper presents a novel subclass of harmonic functions linked to Mittag-Leffler functions, expanding the theoretical framework and properties of such functions.

Findings

Coefficient conditions established for the subclass

Extreme points characterized within the class

Distortion bounds derived for functions in the subclass

Abstract

In the present paper, we introduce a new subclass of harmonic functions in the unit disc U defined by using the generalized Mittag-Leffler type functions. Coefficient conditions, extreme points, distortion bounds, convex combination are studied.

Equations

$$h(z)=z+\sum _{k=2}^{\infty }{a}_k{z}^k,g(z)=\sum _{k=1}^{\infty }{b}_k{z}^k,|b_1|<1.$$

$$E_{\alpha }(z)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (\alpha k+1)},\alpha \in C,R(\alpha )>0,z\in C.$$

$$E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (\beta +\alpha k)},\alpha ,\beta \in C,R(\alpha )>0,R(\beta )>0,z\in C.$$

$$E_{\alpha ,\beta }^{\gamma }(z)=\sum _{k=0}^{\infty }\frac{(\gamma {)}_k{z}^k}{k!\Gamma (\beta +\alpha k)},\alpha ,\beta \in C,R(\alpha )>0,R(\beta )>0,z\in C.$$

$$E_{\alpha ,\beta }^{\gamma ,\delta }(z)=\sum _{k=0}^{\infty }\frac{(\gamma {)}_k{z}^k}{\Gamma (\beta +\alpha k)(\delta {)}_k},$$

$$E_{\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z)=\sum _{k=0}^{\infty }\frac{(\gamma {)}_{qk}{z}^k}{\Gamma (\beta +\alpha k)(\delta {)}_{pk}},$$

$${\Theta }_{\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z)=z\ast E_{\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z)$$

$$=z+\sum _{k=2}^{\infty }\frac{(\gamma {)}_{q(k-1)}}{\Gamma [\beta +\alpha (k-1)](\delta {)}_{p(k-1)}}{z}^k.$$

$${\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^{0}f(z)=f(z)\ast {\Theta }_{\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z)$$

$${\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^{1}f(z)=z[f(z)\ast {\Theta }_{\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z){]}^{{}^{\prime }}$$

$$(m+1){\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^{m+1}f(z)=z[{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m]+m{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m,z\in U.$$

$${\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m=z+\sum _{k=2}^{\infty }\frac{(m+1{)}_{k-1}}{(k-1)!}\frac{(\gamma {)}_{q(k-1)}}{\Gamma [\beta +\alpha (k-1)](\delta {)}_{p(k-1)}}{a}_k{z}^k.$$

$$\Re \left(\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m}{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}\right)>\eta ,$$

$$h(z)=z-\sum _{k=2}^{\infty }{a}_k{z}^k,g_m(z)=(-1{)}^{m-1}\sum _{k=1}^{\infty }{b}_k{z}^k.$$

$$\sum _{k=1}^{\infty }\left(\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|a_k|+\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-(-1{)}^{m-n}\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|b_k|\right)\le 2.$$

$$\ge \sum _{k=1}^{\infty }k|b_k||z{|}^{k-1}\ge |g^{\prime }(z)|.$$

$$-\left|(1+\eta ){\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n(z)-{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m(z)\right|>0.$$

$$\left|(1-\eta ){\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n+{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m\right|-\left|(1+\eta ){\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n-{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m\right|>0.$$

$$=2(1-\eta )|z|$$

$$\left\{1-\sum _{k=2}^{\infty }\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|a_k||z{|}^{k-1}-\sum _{k=1}^{\infty }\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-(-1{)}^{m-n}\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|b_k||z{|}^{k-1}\right\}$$

$$>2(1-\eta )|z|\left\{1-\left(\sum _{k=2}^{\infty }\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|a_k|+\sum _{k=1}^{\infty }\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-(-1{)}^{m-n}\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|b_k|\right)\right\}.$$

$$f(z)=z+\sum _{k=2}^{\infty }\frac{1-\eta }{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{x}_k{z}^k+\sum _{k=1}^{\infty }\frac{1-\eta }{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-(-1{)}^{m-n}\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}\overline{y_k{z}^k},$$

$$\sum _{k=1}^{\infty }\left(\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|a_k|+\frac{{\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-(-1{)}^{m-n}\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n}{1-\eta }|b_k|\right)=1+\sum _{k=2}^{\infty }|x_k|+\sum _{k=1}^{\infty }|y_k|=2.$$

$$\sum _{k=1}^{\infty }\left[\left({\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n\right)|a_k|+\left({\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^m-(-1{)}^{m-n}\eta {\Phi }_{\gamma ,\delta ,q,\alpha ,\beta ,p}^n\right)|b_k|\right]\le 2(1-\eta ).$$

$$(f_m\ast F_m)(z)=f_m(z)\ast F_m(z)=z-\sum _{k=2}^{\infty }|a_k||A_k|z^n+(-1{)}^k\sum _{k=1}^{\infty }|b_k||B_k|{\bar{z}}^k.$$

$$f_{m,j}(z)=z-\sum _{k=2}^{\infty }|a_{k,j}|z^k+(-1{)}^{m-1}\sum _{k=1}^{\infty }|b_{k,j}|{\bar{z}}^k.$$

$$\varpi (z)=\mu f_{m,j}(z)+(1-\mu )f_{m,j}(z),0\le \mu \le 1,$$

$$f_m(z)=\sum _{k=1}^{\infty }\left[X_k{h}_k(z)+Y_k{g}_{m_k}(z)\right],$$

Full text

Keywords: Harmonic functions, hypergeometric functions, Mittag-Leffler type functions.

AMS Mathematics Subject Classification: 30C45

1. Introduction

A continuous complex-valued function $f=u+iv$ defined in a simply complex domain $D$ is said to be harmonic in $D$. In any simply connected domain, we can write $f=h+\bar{g}$, where $h$ and $g$ are analytic in $D$. A necessary and sufficient condition for $f$ to be locally univalent and sense preserving in $D$ is that$|h^{{}^{\prime}}(z)|>|g^{{}^{\prime}}(z)|,z\in D$.

Clunie and Sheil-Small [1] introduced a class $S{H}$ of complex valued harmonic maps f which are univalent and sense-preserving in the open unit disk $U=\{{z:|z|<1}\}$ and assume a normalized representation $f=h+\bar{g}$ where $f(0)=f_{z}(0)-1=0$. Then for $f=h+\bar{g}\in S{H}$ we may express the analytic functions $h$ and $g$ as

[TABLE]

In 1984, Clunie[1] and Sheil-Small [2] investigated the class $S{H}$ as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on $S{H}$ and its subclasses. Connectivity of geometric functions and hypergeometric functions with harmonic functions is seen through some of the these papers [[3][4], [5], [6],[7]]. The Mittag-Leffler and generalized Mittag-Leffler type functions was first introduced by the swedish mathematician G. M. Mittag-Leffler [8] and also studied by Wiman [9]. It is a special function of $z\in C$ which depends on the complex parameter $\alpha$ and is defined by the power series

[TABLE]

A first generalization of $E_{\alpha}(z)$ introduced by Wiman [9], is the two-parametric M-L function of $z\in C$, defined by the series

[TABLE]

1971 Prabhakar [10] introduced a three-parametric generalization of $\psi^{\gamma}_{\alpha,\beta}(z)$ defined in (3) as a kernel of certain fractional differential equations in terms of the series

[TABLE]

Due to its integral representation $E^{\gamma}_{\alpha,\beta}(z)$ is considered as a special case of Fox’s H-function as well as of Wright’s generalized hypergeometric ${}_{p}\Psi_{q}$, so called Fox-Wright psi function of $z\in C$. Later, Salim [11] introduced the function in the form $\psi^{\gamma}_{\alpha,\beta}(z)$ in the following form

[TABLE]

where $\alpha,\beta,\gamma,\delta\in C,\min(R(\alpha),R(\beta)>0,R(\gamma),R(\delta)>0),z\in C.$ Recently, Salim and Faraj [12] introduced a new generalization of Mittag-Leffler function associated with Weyl fractional integral and differential operators as follow

[TABLE]

where $\alpha,\beta,\gamma,\delta\in C,\min(R(\alpha),R(\beta)>0,R(\gamma),R(\delta)>0),z\in C$, with $q,p\in\mathbb{R}_{+}$, $q\leq\Re(\alpha)+p$, and $(\gamma)_{pn}$ denotes an extended variant of the Pochhammer symbol, defined by $(\gamma)_{qn}=\Gamma(\gamma+qn)/\Gamma(\gamma)$.

Corresponding to $E^{\gamma,\delta,q}_{\alpha,\beta,p}(z)$, we define the function $\Theta^{\gamma,\delta,q}_{\alpha,\beta,p}(z)$ by

[TABLE]

[TABLE]

Now, for $f\in A,m\in\mathbb{N}$, we define the following differential operator: $\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p})f:A\rightarrow A$ by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus it is obvious to see from above that

[TABLE]

Note that, when $\alpha=0,\beta=\gamma=\delta=1$, we get Ruscheweyh Operator [13].

Involving the generalized Mittag-Leffler function as defined in (6), for $0\leq\eta<1,m\in\mathbb{N},n\in\mathbb{N}_{0},m>n$ and $z\in U$, we let $SH(m,n,\eta)$ denote the family of harmonic functions f of the form (1) such that

[TABLE]

where $\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}=\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}h(z)+(-1)^{m}\overline{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}}g(z).$

We let the subclass $\overline{S}H(m,n,\eta)$ consist of harmonic functions $f_{m}=h+\overline{g}_{m}$ in $\overline{S}H(m,n,\eta)$ so that $h$ and $g_{m}$ are of the form

[TABLE]

The class $\overline{S}H(m,n,\eta)$ includes a variety of well-known subclasses of $SH$. The object of this paper is to examine some generalized Mittag-Leffler function inequalities as a necessary and sufficient conditions for univalent harmonic analytic functions associated with certain generalized Mittag-Leffler function to be in the function class $SH(m,n,\eta)$. The coefficient condition for the function class $SH(m,n,\eta)$ is given. Furthermore, we determine extreme points, a distortion theorem, convolution conditions and convex combinations for the functions $f$ in $\overline{S}H(m,n,\eta)$.

2. Coefficient bound

We begin with a sufficient coefficient condition for functions $f$ in $SH(m,n,\eta)$.

Theorem 2.1**.**

Let $f=h+\overline{g}$ be given by (1). If

[TABLE]

Proof. If $z_{1}\neq z_{2}$ , then

$\left|\frac{f(z_{1})-f(z_{2})}{h(z_{1})-h(z_{2})}\right|\geq 1-\left|\frac{g(z_{1})-g(z_{2})}{h(z_{1})-h(z_{2})}\right|$

$=1-\left|\frac{\sum^{\infty}_{k=1}b_{k}(z^{k}_{1}-z^{k}_{2})}{(z_{1}-z_{2})+\sum^{\infty}_{k=2}a_{k}(z^{k}_{1}-z^{k}_{2})}\right|>1-\frac{\sum^{\infty}_{k=1}k|b_{k}|}{1-\sum^{\infty}_{k=2}k|a_{k}|}$

$\geq 1-\frac{\frac{\sum^{\infty}_{k=1}\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}|b_{k}|}{1-\sum^{\infty}_{k=2}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}|a_{k}|}\geq 0,$

which proves univalence. Note that $f$ is sense-preserving in $U$. This is because

$|h^{{}^{\prime}}(z)|=1-\sum^{\infty}_{k=2}k|a_{k}||z|^{k-1}$

$>1-\sum^{\infty}_{k=2}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}|a_{k}|$

$\geq\sum^{\infty}_{k=1}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}|b_{k}|$

$>\sum^{\infty}_{k=1}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}|b_{k}||z^{k}|$

[TABLE]

Using the fact that $\Re(w)>\eta$ if and only if $|1-\eta+w|\geq|1+\eta-w|$, it suffices to show that

$\left|(1-\eta)\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}(z)+\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}(z)\right|$

[TABLE]

Substituting the value of $\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}(z)$ and $\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}(z)$ in (11) yields, by (9), that

[TABLE]

$=|(2-\eta)z+\sum^{\infty}_{k=2}[\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}(1-\eta)+\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}]a_{k}z^{k}$

$+(-1)^{n}\sum^{\infty}_{k=1}[\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}(1-\eta)+(-1)^{m-n}\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}]\overline{b_{k}z^{k}}$

$-|-\eta z+\sum^{\infty}_{k=2}[\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(1+\eta)\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}]a_{k}z^{k}|$

$-(-1)^{n}\sum^{\infty}_{k=1}[(-1)^{m-n}\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(1+\eta)\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}]\overline{b_{k}z^{k}}|$

$\geq 2(1-\eta)|z|-\sum^{\infty}_{k=2}2[\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}]|a_{k}|z|^{k}$

$-\sum^{\infty}_{k=1}[(1+\eta)\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}+(-1)^{m-n}\eta\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}]|b_{k}||z|^{k}$

$-\sum^{\infty}_{k=1}|[(-1)^{m-n}\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(1+\eta)\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}]|b_{k}||z|^{k}$

[TABLE]

[TABLE]

[TABLE]

This last expressions is non-negative by (10), and so the proof is complete.

The harmonic function

[TABLE]

where $m\in\mathbb{N},n\in\mathbb{N}_{0}$ and $\sum^{\infty}_{k=2}|x_{k}|+\sum^{\infty}_{k=1}|y_{k}|=1$ shows that the coefficient bound given by (10) is sharp. The functions $f$ of the form (13) are $f\in SH(m,n,\eta)$, because

[TABLE]

In the following theorem, it is shown that the condition (10) is also necessary for functions $f_{m}=h+\overline{g_{m}}$, where $h$ and $g_{m}$ are of the form (9).

Theorem 2.2**.**

Let $f_{m}=h+\overline{g_{m}}$ be given by (9). Then $f_{m}\in\overline{S}H(m,n,\eta)$ If and only if

[TABLE]

Proof. Since $\overline{S}H(m,n,\eta)\subset\overline{S}H(m,n,\eta)$, we only need to prove the ”only if ” part of Theorem 2.2. To this end, for functions $f$ of the form (9), we notice that the condition (8) is equivalent to

$\Re\left\{\frac{(1-\eta)z-\sum_{k=2}^{\infty}\left(\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}\right)a_{k}z^{k}+(-1)^{2m-1}\sum_{k=1}^{\infty}\left(\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}\right)b_{k}\overline{z}^{k}}{z-\sum_{k=2}^{\infty}\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}a_{k}z^{k}+(-1)^{m+n-1}\sum_{k=1}^{\infty}\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}b_{k}\overline{z}^{k}}\right\}\geq 0.$

[TABLE]

The above condition (15) must hold for all values of $z$ on the positive real axes, where, $0\leq|z|=\mu<1,$ we have

$\frac{1-\eta-\sum_{k=2}^{\infty}\left(\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}\right)a_{k}\mu^{k-1}-\sum_{k=1}^{\infty}\left(\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}\right)b_{k}\mu^{k-1}}{1-\sum_{k=2}^{\infty}\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}a_{k}\mu^{k-1}+(-1)^{m-n}\sum_{k=1}^{\infty}\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}b_{k}\mu^{k-1}}\geq 0.$

[TABLE]

If the condition (14) does not hold, then the numerator in (16) is negative for sufficiently close to 1. Hence there exists a $z_{0}=\mu_{0}$ in $(0,1)$ for which the quotient in (16) is negative. This contradicts the condition for $f_{j}\in\overline{S}H(m,n,\eta)$ and so the proof is complete.

3. Distortion bounds

In this section, we obtain distortion bounds for functions $f$ in $\overline{S}H(m,n,\eta).$

Theorem 3.1**.**

Let $f_{j}\in\overline{S}H(m,n,\eta)$, then for $|z|<1$, we have

$|f_{k}(z)|\geq(1-|b_{1}|)\mu-\frac{1}{\Upsilon_{2}}\left(\frac{(1-\eta)}{[\Upsilon_{2}]^{m-n}-\eta}-\frac{1-(-1)^{m-n}\eta}{[\Upsilon_{2}]^{m-n}-\eta}|b_{1}|\right)\mu^{2}$**

and

$|f_{k}(z)|\geq(1-|b_{1}|)\mu-\frac{1}{\Upsilon_{2}}\left(\frac{(1-\eta)}{[\Upsilon_{2}]^{m-n}-\eta}-\frac{1-(-1)^{m-n}\eta}{[\Upsilon_{2}]^{m-n}-\eta}|b_{1}|\right)\mu^{2},$**

where $\Upsilon_{2}=(m+1)\left(\frac{(\gamma)_{q}}{\Gamma[\beta+\alpha](\delta)_{p}}\right).$

Proof. We only prove the left-hand inequality. The proof for the right-hand inequality is similar and is thus omitted. Let $f_{m}\in\overline{S}H_{\alpha,\beta,p,n}^{\gamma,\delta,q,m}(\eta)$. Taking the absolute value of $f_{m}$, we have

$|f_{m}(z)|=|z-\sum_{k=2}^{\infty}a_{k}z^{k}+(-1)^{k}\sum_{k=1}^{\infty}b_{k}\overline{z}^{k}|$

$\leq(1+|b_{1}|)\mu+\sum_{k=2}^{\infty}(|a_{k}|+|b_{k}|)\mu^{n}$

$\leq(1+|b_{1}|)\mu+\sum_{k=2}^{\infty}(|a_{k}|+|b_{k}|)\mu^{2}$

$=(1+|b_{1})\mu+\frac{1-\eta}{[\Upsilon_{2}]^{m}([\Upsilon_{2}]^{m-n}-\eta)}\sum^{\infty}_{k=2}(\frac{\Upsilon_{2}]^{m}([\Upsilon_{2}]^{m-n}-\eta)}{1-\gamma}(|a_{k}|+|b_{k}|)\mu^{2}$

$\leq(1+|b_{1})\mu+\frac{(1-\eta)\mu^{2}}{[\Upsilon_{2}]^{m}([\Upsilon_{2}]^{m-n}-\eta)}\sum^{\infty}_{k=2}\left(\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}a_{k}+\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}b_{k}\right)$

$\leq(1+|b_{1})\mu+\frac{(1-\eta)\mu^{2}}{[\Upsilon_{2}]^{m}([\Upsilon_{2}]^{m-n}-\eta)}\left(\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}a_{k}+\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}b_{k}\right)\mu^{2}$.

The following covering result follows from the left hand inequality in Theorem 3.1.

4. Convolution, convex combinations and extreme points

In this section, we show the class $SH(m,n,\eta)$ is invariant under convolution and convex combination.

For harmonic functions $f$ of the form

$f_{m}(z)=z-\sum_{k=2}^{\infty}|a_{k}|z^{k}+(-1)^{m-1}\sum_{k=1}^{\infty}|b_{k}|\bar{z}^{k}$

and

$F_{m}(z)=z-\sum_{k=2}^{\infty}|A_{k}|z^{k}+(-1)^{m-1}\sum_{k=1}^{\infty}|B_{k}|\bar{z}^{k},$

we define the convolution of $f_{m}(z)$ and $F_{m}(z)$ as

[TABLE]

Theorem 4.1**.**

For $0\leq\rho\leq\eta<1$, let $f_{m}\in\overline{S}H(m,n,\eta)$ and $F_{m}\in\overline{S}(m,n,\rho)$. Then the convolution

$f_{m}*F_{m}\in\overline{S}H(m,n,\eta)\subset\overline{S}(m,n,\rho).$**

*Proof. *Then the convolution $f_{m}*F_{m}$ is given by (17). We wish to show that the coefficients of $f_{m}*F_{m}$ satisfy the required condition given in Theorem 4.1. For $F_{m}\in\overline{S}H(m,n,\rho)$, we note that $|A_{k}|\leq 1$ and $|B_{k}|\leq 1$. Now, for the convolution function $f_{m}*F_{m}$, we obtain

$\sum^{\infty}_{k=2}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\rho\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\rho}a_{k}A_{k}+\sum^{\infty}_{k=1}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\rho\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\rho}b_{k}B_{k}$

$\leq\sum^{\infty}_{k=2}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\rho\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\rho}a_{k}+\sum^{\infty}_{k=1}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\rho\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\rho}b_{k}$

$\leq\sum^{\infty}_{k=2}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\mu\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\mu}a_{k}+\sum^{\infty}_{k=1}\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\mu\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\mu}b_{k}\leq 1.$

Since $0\leq\rho\leq\eta<1$, and $f_{m}\in\overline{S}H(m,n,\eta)$, then $f_{m}*F_{m}\in\overline{S}H(m,n,\eta)\subset\overline{S}H(m,n,\rho)$.

Next we discuss the convex combinations of the class $\overline{S}H(m,n,\eta)$.

Theorem 4.2**.**

The family $\overline{S}H(m,n,\eta)$ is closed under convex combination.

Proof. For $i=1,2,...$, suppose that $f_{m,j}\in\overline{S}H(m,n,\eta)$, where

[TABLE]

Then by Theorem 2.1,

$\sum^{\infty}_{k=1}\left(\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}a_{k,j}+\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}b_{k,j}\right)\leq 2.$

[TABLE]

For $\sum^{\infty}_{j=1}t_{j}=1$, $0\leq t_{j}<1$, the convex combination of $f_{m,j}$ may be written as

$\sum^{\infty}_{j=1}t_{j}f_{m,j}(z)=z-\sum^{\infty}_{k=2}\left(\sum^{\infty}_{j=1}t_{j}|a_{k,j}|\right)z^{k}+(-1)^{m-1}\sum^{\infty}_{k=1}\left(\sum^{\infty}_{j=1}t_{j}|b_{k,j}|\right)\bar{z}^{k}.$

Then by (19),

$\sum^{\infty}_{k=1}\left(\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}\sum^{\infty}_{j=1}t_{j}a_{k,j}+\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}\sum^{\infty}_{j=1}t_{j}b_{k,j}\right)$

$=\sum^{\infty}_{i=1}t_{i}\left\{\sum^{\infty}_{k=1}\left(\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}a_{k,j}+\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}b_{k,j}\right)\right\}$

$\leq 2\sum^{\infty}_{j=1}t_{j}=2$,

and therefore

$\sum^{\infty}_{j=1}t_{j}f_{m,j}(z)\in\overline{S}H(m,n,\eta).$

Corollary 4.3**.**

The class $\overline{S}H(m,n,\eta)$ is closed under convex linear combinations.

Proof. Let the functions $f_{m,j}(z)\ (j=1,2...,m)$ defined by (18) be in the class $\overline{S}H(m,n,\eta).$ Then the function $\varpi(z)$ defined by

[TABLE]

is in the class $\overline{S}H(m,n,\eta)$.

Next we determine the extreme points of closed convex hulls of $\overline{S}H(m,n,\eta)$, denoted by clco $\overline{S}H(m,n,\eta).$

Theorem 4.4**.**

Let $f_{m}$ be given by (7). Then $f_{m}\in\overline{S}H(m,n,\eta)$ if and only if

[TABLE]

where

$h_{1}(z)=1,\ h_{k}(z)=z-\frac{1-\eta}{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}z^{k},(k=2,...),$

$g_{m_{k}}(z)=z+(-1)^{m-1}\frac{1-\eta}{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}\bar{z}^{n},(k=1,2,...),$

$X_{1}=1-\sum^{\infty}_{k=2}(X_{k}+Y_{k})\geq 0,\ X_{k}\geq 0,\ Y_{k}\geq 0$. In particular, the extreme points of $\overline{S}H(m,n,\eta)$ are $\left\{h_{k}\right\}\ and\ \left\{g_{m_{k}}\right\}$.

Proof. For functions $f_{m}$ of the form (21) we have

$f_{m}(z)=\sum^{\infty}_{k=1}\left[X_{k}h_{k}(z)+Y_{k}g_{m_{k}}(z)\right]$

$=\sum^{\infty}_{k=1}(X_{k}+Y_{k})z-\sum^{\infty}_{k=2}\frac{1-\eta}{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}X_{k}z^{k}$

$+(-1)^{m-1}\sum^{\infty}_{n=1}\frac{1-\eta}{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}Y_{k}\bar{z}^{k}.$

Then

$\sum^{\infty}_{k=2}\left(\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}\right)\frac{1-\eta}{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}X_{k}$

$\sum^{\infty}_{k=1}\left(\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}\right)\frac{1-\eta}{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}Y_{k}.$

$=\sum^{\infty}_{n=2}X_{n}+\sum^{\infty}_{n=1}Y_{n}=1-X_{1}\leq 1,$

[TABLE]

and so $f_{m}\in clco\ \overline{S}H(m,n,\eta).$

Conversely, suppose that $f_{m}\in clco\ \overline{S}H(m,n,\eta).$ Setting

$X_{k}=\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}a_{k},\ 0\leq X_{k}\leq 1,k=2,...\ ,$

$Y_{k}=\frac{\Phi^{m}_{\gamma,\delta,q,\alpha,\beta,p}-(-1)^{m-n}\eta\Phi^{n}_{\gamma,\delta,q,\alpha,\beta,p}}{1-\eta}b_{k},\ 0\leq Y_{k}\leq 1,k=1,2,...\ ,$

and $X_{1}=1-\sum^{\infty}_{k=2}X_{k}+\sum^{\infty}_{k=1}Y_{k}$ , and note that, by Theorem 2.1, $x_{1}\geq 0$. Consequently, we obtain

$f_{m}(z)=\sum^{\infty}_{k=1}\left(h_{k}(z)X_{k}+g_{m_{k}}(z)Y_{k}\right),$ as required.

Using Corollary 4.3 we have $clco\ \overline{S}H(m,n,\eta)=\overline{G}_{S}H(m,n,\eta)$. Then the statement of Theorem 4.4 is true for $f\in\overline{G}_{S}H(m,n,\rho).$