Subclass of Analytic Functions Connected with Double Zeta Function

Abstract

In this survey-cum-expository work, we primarily seek to study many families of the renowned Hurwitz–Lerch Zeta mapping, including the so-called generalized Hurwitz–Lerch Zeta mappings. The purpose of this study is to examine a new subclass of Hurwitz–Lerch Zeta mappings with negative coefficients in the unit disc $U=\{z\in \mathbb{C}:|z|<1\}.$ We explore fundamental characteristics of the defined class, such as coefficient inequality, neighborhoods, partial sums, and integral means properties.

1. Introduction

Let A indicate the class of all mappings $\mathsf{\Lambda }\left(\omega \right)$ of the form

$$\mathsf{\Lambda }\left(\omega \right)=\omega +\sum _{\ell =2}^{\infty }{\mathsf{\varrho }}_{\ell }{\omega }^{\ell },$$

(1)

which are analytic in the open unit disc $U=\{\omega \in \mathbb{C}:|\omega |<1\}.$ Let S be the subclass of A consisting of univalent mappings and fulfill the following usual normalization condition

$$\mathsf{\Lambda }\left(0\right)={\mathsf{\Lambda }}^{\prime }\left(0\right)-1=0.$$

We indicate by S the subclass of A consisting of mappings $\mathsf{\Lambda }\left(\omega \right)$ which are all univalent in $U.$ A mapping $\mathsf{\Lambda }\in A$ is a starlike mapping of the order $\xi ,0\le \xi <1,$ if it fulfills

$$\Re \left\{\frac{\omega {\mathsf{\Lambda }}^{\prime }\left(\omega \right)}{\mathsf{\Lambda }\left(\omega \right)}\right\}>\xi ,(\omega \in U).$$

(2)

We indicate this class with $S^{\ast }\left(\xi \right).$ A mapping $\mathsf{\Lambda }\in A$ is a convex mapping of the order $\xi ,0\le \xi <1,$ if it fulfills

$$\Re \left\{1+\frac{\omega {\mathsf{\Lambda }}^{″}\left(\omega \right)}{{\mathsf{\Lambda }}^{\prime }\left(\omega \right)}\right\}>\xi ,(\omega \in U).$$

(3)

We indicate this class with $K\left(\xi \right).$ Note that $S^{\ast }\left(0\right)=S^{\ast }$ and $K\left(0\right)=K$ are the usual classes of starlike and convex mappings in $U,$ respectively. For $\mathsf{\Lambda }\in A$ given by (1) and $g\left(\omega \right)$ given by

$$g\left(\omega \right)=\omega +\sum _{\ell =2}^{\infty }{b}_{\ell }{\omega }^{\ell }$$

(4)

their convolution (or Hadamard product), indicated by $(\mathsf{\Lambda }\ast g),$ is defined as

$$(\mathsf{\Lambda }\ast g)\left(\omega \right)=\omega +\sum _{\ell =2}^{\infty }{\mathsf{\varrho }}_{\ell }{b}_{\ell }{\omega }^{\ell }=(g\ast \mathsf{\Lambda })\left(\omega \right),(\omega \in U).$$

(5)

Note that $\mathsf{\Lambda }\ast g\in A.$

Let T indicate the class of mappings analytic in A that are of the form

$$\mathsf{\Lambda }\left(\omega \right)=\omega -\sum _{\ell =2}^{\infty }{\mathsf{\varrho }}_{\ell }{\omega }^{\ell },({\mathsf{\varrho }}_{\ell }\ge 0,\omega \in U)$$

(6)

and let $T^{\ast }\left(\xi \right)=T\cap S^{\ast }\left(\xi \right),C\left(\xi \right)=T\cap K\left(\xi \right).$ The class $T^{\ast }\left(\xi \right)$ and allied classes possess some interesting properties and have been carefully examined by Silverman [1].

A mapping $\mathsf{\Lambda }\in A$ is said to be in $\aleph -US\left(j\right),$ the class of $\aleph -$uniformly starlike mappings of order $j, 0\le j<1,$ if it fulfills the condition

$$\Re \left\{\frac{\omega {\mathsf{\Lambda }}^{\prime }\left(\omega \right)}{\mathsf{\Lambda }\left(\omega \right)}\right\}>\aleph \left|\frac{\omega {\mathsf{\Lambda }}^{\prime }\left(\omega \right)}{\mathsf{\Lambda }\left(\omega \right)}-1\right|+j,(\aleph \ge 0),$$

(7)

and a mapping $\mathsf{\Lambda }\in A$ is said to be in $\aleph -UC\left(j\right),$ the class of $\aleph -$uniformly convex mappings of order $j, 0\le j<1,$ if it fulfills the condition

$$\Re \left\{1+\frac{\omega {\mathsf{\Lambda }}^{″}\left(\omega \right)}{{\mathsf{\Lambda }}^{\prime }\left(\omega \right)}\right\}>\aleph \left|\frac{\omega {\mathsf{\Lambda }}^{″}\left(\omega \right)}{{\mathsf{\Lambda }}^{\prime }\left(\omega \right)}\right|+j,(\aleph \ge 0).$$

(8)

Goodman [2] described uniformly starlike and uniformly convex mappings originally, and a group of authors have since explored them.

The class $S_s$ of starlike mappings with regard to symmetric points was defined by Sakaguchi in [3], as shown in:

Let $\mathsf{\Lambda }\in A.$ Then, $\mathsf{\Lambda }$ is said to be starlike with respect to symmetric points in $U⟺$

$$\Re \left\{\frac{2\omega {\mathsf{\Lambda }}^{\prime }\left(\omega \right)}{\mathsf{\Lambda }\left(\omega \right)-\mathsf{\Lambda }(-\omega )}\right\}>0,(\omega \in U).$$

Recently, Owa et al. [4] defined the class $S_s(\alpha ,ħ)$ as follows:

$$\Re \left\{\frac{(1-ħ)\omega {\mathsf{\Lambda }}^{\prime }\left(\omega \right)}{\mathsf{\Lambda }\left(\omega \right)-\mathsf{\Lambda }(ħ\omega )}\right\}>\alpha ,(\omega \in U),$$

where $0\le \alpha <1,|ħ|\le 1,ħ\ne 1.$ Note that $S_s(0,-1)=S_s$ and $S_s(\alpha ,-1)=S_s\left(\alpha \right)$ are called Sakaguchi mapping of order $\alpha .$

The study of operators is crucial to the understanding of geometric mapping theory and its related subjects.

The assessment of numerous series families connected to the Riemann and Hurwitz zeta mappings, as well as their generalisations and extensions, such as the Hurwitz–Lerch zeta mapping, have attracted a lot of attention in recent years. These mappings ascend naturally in many branches of analytic mapping theory and their studies have plentiful important applications in mathematics [5]. As a overview of both Riemann and Hurwitz zeta mappings, the so-called Hurwitz–Lerch zeta mapping is defined in [6]. Hurwitz–Lerch Zeta mapping $\mathsf{\Psi }(\omega ,s,\mathsf{\varrho })$ is defined in [7] given by

$$\mathsf{\Psi }(\omega ,s,\mathsf{\varrho }):=\sum _{\ell =0}^{\infty }\frac{{\omega }^{\ell }}{{(\ell +\mathsf{\varrho })}^s}$$

(9)

$(\mathsf{\varrho }\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}; s\in \mathbb{C};\mathfrak{R}\left(s\right)>1)$ and $\left|\omega \right|<1$ where ${\mathbb{Z}}_0^{-}:=\mathbb{Z}\backslash \mathbb{N},(\mathbb{Z}:=\left\{\pm 0,\pm 1,\pm 2,\pm 3,\cdots \right\}).$ It is clear that $\mathsf{\Psi }$ is an analytic mapping in both variables s and $\omega$ in a suitable region, and it eases to the ordinary Lerch zeta mapping $\omega =2\pi i\lambda .$ In addition, $\mathsf{\Psi }$ yields the following known result [6]:

$$\mathsf{\Psi }(\omega ,1,\mathsf{\varrho })={\mathsf{\varrho }}^{-1}{}_2{F}_1(\mathsf{\varrho },1;\mathsf{\varrho }+1,\omega ),$$

where ${}_2{F}_1$ is the Gaussian hypergeometric mapping. Several interesting properties and characteristics of the Hurwitz–Lerch Zeta mapping $\mathsf{\Psi }(\omega ,s,\mathsf{\varrho })$ can be found in the recent investigations by Choi and Srivastava [8], and (also see [9]) the reference stated therein.

The double zeta mapping of Barnes [10] (and also see [11])

$$\zeta (x,\mathsf{\varrho },\sigma )=\sum _{\ell =0}^{\infty }\sum _{m=0}^{\infty }{(m+\mathsf{\varrho }+\sigma \ell )}^{-x},$$

where $\mathsf{\varrho }\ne 0$ and $\sigma$ is a non zero complex number with $\left|arg\right(\sigma \left)\right|<\pi .$ Bin- Saad [12] posed a generalized double zeta mapping of the form

$${\zeta }_{\sigma }^{\nu }(\omega ,s,\mathsf{\varrho })=\sum _{\ell =0}^{\infty }{\left(\nu \right)}_{\ell }\mathsf{\Psi }(\omega ,s,\mathsf{\varrho }+\ell \sigma )\frac{{\omega }^{\ell }}{\ell !}$$

where $\sigma \in \mathbb{C}\backslash \left\{0\right\};\nu \in \mathbb{C}\backslash {\mathbb{Z}}_0^{-};\mathsf{\varrho }\in \mathbb{C}\backslash \left\{-(m+\sigma \ell )\right\},\ell ,m\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\},\left|s\right|<1;\left|\omega \right|<1$, and $\mathsf{\Psi }$ is the Hurwitz–Lerch zeta mapping distinct by (9), and ${\left(\nu \right)}_{\ell }$ is the Pochhammer symbol defined by

$${\left(\nu \right)}_{\ell }=\left\{\begin{array}{cc}1,\hfill & \ell =0\hfill \\ \nu (\nu +1)(\nu +2)\cdots (\nu +\ell -1),\hfill & \ell \in \mathbb{N}.\hfill \end{array}\right.$$

(10)

In this work, using the Hadamard product or the convolution product of a generalized Hurwitz–Lerch zeta mapping in [11] is defined as follows:

$${\mathsf{\Theta }}_{\ell }(\omega ,s,\mathsf{\varrho })=\frac{\mathsf{\Psi }(\omega ,s,\mathsf{\varrho }+\ell \sigma )}{\mathsf{\Psi }(\omega ,s,\ell )},\ell \in {\mathbb{N}}_0.$$

(11)

It is clear that ${\mathsf{\Theta }}_0(\omega ,s,\mathsf{\varrho })=1.$ Now, we consider the mapping

$${\mathsf{{\rm Y}}}_{\nu }(\omega ,s,\mathsf{\varrho })=\sum _{\ell =0}^{\infty }\frac{{\left(\nu \right)}_{\ell }}{\ell !}{\mathsf{\Theta }}_{\ell }(\omega ,s,\mathsf{\varrho }){\omega }^{\ell },$$

(12)

which implies

$$\omega {\mathsf{{\rm Y}}}_{\nu }(\omega ,s,\mathsf{\varrho })=\omega +\sum _{\ell =2}^{\infty }\frac{{\left(\nu \right)}_{\ell -1}}{(\ell -1)!}{\mathsf{\Theta }}_{\ell -1}(\omega ,s,\mathsf{\varrho }){\omega }^{\ell }.$$

Thus,

$$\omega {\mathsf{{\rm Y}}}_{\nu }(\omega ,s,\mathsf{\varrho })\ast {\left(\omega {\mathsf{{\rm Y}}}_{\nu }(\omega ,s,\mathsf{\varrho })\right)}^{-1}=\frac{\omega }{{(1-\omega )}^{\delta }}=\omega +\sum _{\ell =2}^{\infty }\frac{{\left(\nu \right)}_{\ell -1}}{(\ell -1)!}{\omega }^{\ell },\delta >-1$$

possesses a linear operator

$${\mathfrak{J}}_{\nu }^{\delta }(\omega ,s,\mathsf{\varrho })\mathsf{\Lambda }\left(\omega \right)={\left(\omega {\mathsf{{\rm Y}}}_{\nu }(\omega ,s,\mathsf{\varrho })\right)}^{-1}\ast \mathsf{\Lambda }\left(\omega \right)=\omega +\sum _{\ell =2}^{\infty }\frac{{\left(\delta \right)}_{\ell -1}}{{\left(\nu \right)}_{\ell -1}{\mathsf{\Theta }}_{\ell -1}(\omega ,s,\mathsf{\varrho })}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell },$$

(13)

where $\sigma \in \mathbb{C}\backslash \left\{0\right\};\nu \in \mathbb{C}\backslash {\mathbb{Z}}_0^{-};a\in \mathbb{C}\backslash \left\{-(m+\sigma \ell )\right\},\ell ,m\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\},\left|s\right|<1;\left|\omega \right|<1$ and ${\mathsf{\Theta }}_{\ell }(\omega ,s,\mathsf{\varrho })$ is defined in (11). It is clear that

$${\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)={\mathfrak{J}}_{\nu }^{\delta }(\omega ,s,\mathsf{\varrho })\mathsf{\Lambda }\left(\omega \right)=\omega +\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }{\mathsf{\varrho }}_{\ell }{\omega }^{\ell },$$

(14)

where

$${\mathsf{\Psi }}_{\ell }=\frac{{\left(\delta \right)}_{\ell -1}}{{\left(\nu \right)}_{\ell -1}{\mathsf{\Theta }}_{\ell -1}(\omega ,s,\mathsf{\varrho })}.$$

Now, by making use of the Hurwitz–Lerch zeta operator ${\mathfrak{J}}_{\nu }^{\delta }f,$ we define a new subclass of mappings motivated by the recent work of Thirupathi Reddy and Venkateswarlu [13] and Venkateswarlu et al. [14].

Definition 1.

A mapping $\mathsf{\Lambda }\in A$ is said to be in the class $\aleph -U{S}_s(\nu ,\delta ,j,ħ)$ if for all $\omega \in U$

$$\Re \left\{\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}\right\}\ge \aleph \left|\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}-1\right|+j,$$

for $\aleph \ge 0,|ħ|\le 1,ħ\ne 1,0\le j<1.$

Furthermore, we say that a mapping $\mathsf{\Lambda }\in \aleph -U{S}_s(\nu ,\delta ,j,ħ)$ is in the subclass $\aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)$ if $\mathsf{\Lambda }\left(\omega \right)$ is of the following form (6).

The aim of the present paper is to study the coefficient bounds, partial sums, certain neighborhood results and integral means property of the class $\aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ).$ Firstly, we shall need the following Lemmas [15].

Lemma 1.

Let w be a complex number. Then,

$$\Re \left(w\right)\ge \alpha ⟺|w-(1+\alpha \left)\right|\le |w+(1-\alpha \left)\right|.$$

Lemma 2.

Let w be a complex number and $\alpha ,j$ be real numbers. Then,

$$\Re \left(w\right)>\alpha |w-1|+j⟺\Re \{w(1+\alpha e^{i\theta })-\alpha e^{i\theta }\}>j,-\pi \le \theta <\pi .$$

2. Coefficient Bounds

Theorem 1.

Let $\mathsf{\Lambda }\in T.$ Then, $\mathsf{\Lambda }\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)⟺$

$$\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }|\ell (\aleph +1)-{\mathsf{\Lambda }}_{\ell }(\aleph +j)|{\mathsf{\varrho }}_{\ell }\le 1-j,$$

(15)

where $\aleph \ge 0,|ħ|\le 1,ħ\ne 1,0\le j<1$ and ${\mathsf{\Lambda }}_{\ell }=1+ħ+\cdots +{ħ}^{\ell -1}.$

The result is sharp for the mapping $\mathsf{\Lambda }\left(\omega \right)$ given by

$$\mathsf{\Lambda }\left(\omega \right)=\omega -\frac{1-j}{{\mathsf{\Psi }}_{\ell }|\ell (\aleph +1)-{\mathsf{\Lambda }}_{\ell }(\aleph +j)|}{\omega }^{\ell }.$$

Proof. 

By Definition 1, we obtain

$$\Re \left\{\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}\right\}\ge \aleph \left|\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}-1\right|+j.$$

Then, by Lemma 2, we have

$$\Re \left\{\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}(1+\aleph e^{i\theta })-\aleph e^{i\theta }\right\}\ge j,-\pi \le \theta <\pi$$

or equivalently

$$\Re \left\{\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}-\frac{\aleph e^{i\theta }\left[{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right]}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}\right\}\ge j.$$

(16)

Let $H\left(\omega \right)=(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })-\aleph e^{i\theta }\left[{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right]$ and $E\left(\omega \right)={\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega ).$

By Lemma 1, (16) is equivalent to

$$\left|H\right(\omega )+(1-j\left)E\right(\omega \left)\right|\ge \left|H\right(\omega )-(1+j\left)E\right(\omega \left)\right|,\mathrm{for}0\le j<1.$$

However,

$$\begin{array}{cc}\hfill \left|H\right(\omega )+(1-j\left)E\right(\omega \left)\right|=& |(1-ħ)\{(2-j)\omega -\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }(\ell +{\mathsf{\Lambda }}_{\ell }(1-j)){\mathsf{\varrho }}_{\ell }{\omega }^{\ell }\hfill \\ \hfill & -\aleph e^{i\theta }\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }(\ell -{\mathsf{\Lambda }}_{\ell }){\mathsf{\varrho }}_{\ell }{\omega }^{\ell }\}|\hfill \\ \hfill \ge & |1-ħ|\left\{(2-j)\right|\omega |-\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }|\ell +{\mathsf{\Lambda }}_{\ell }(1-j)|{\mathsf{\varrho }}_{\ell }\left|{\omega }^{\ell }\right|\hfill \\ \hfill & -\aleph \sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }|\ell -{\mathsf{\Lambda }}_{\ell }|{\mathsf{\varrho }}_{\ell }\left|{\omega }^{\ell }\right|\}.\hfill \end{array}$$

In addition,

$$\begin{array}{cc}\hfill \left|H\right(\omega )-(1+j\left)E\right(\omega \left)\right|=& \left|(1-ħ)\right\{-j\omega -\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }(\ell -{\mathsf{\Lambda }}_{\ell }(1+j)){\mathsf{\varrho }}_{\ell }{\omega }^{\ell }\hfill \\ \hfill & -\aleph e^{i\theta }\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }(\ell -{\mathsf{\Lambda }}_{\ell }){\mathsf{\varrho }}_{\ell }{\omega }^{\ell }\}|\hfill \\ \hfill \le & |1-ħ|\left\{j\right|\omega |+\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }|\ell -{\mathsf{\Lambda }}_{\ell }(1+j)|{\mathsf{\varrho }}_{\ell }\left|{\omega }^{\ell }\right|\hfill \\ \hfill & +\aleph \sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }|\ell -{\mathsf{\Lambda }}_{\ell }|{\mathsf{\varrho }}_{\ell }\left|{\omega }^{\ell }\right|\}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & \left|H\right(\omega )+(1-j\left)E\right(\omega \left)\right|-\left|H\right(\omega )-(1+j\left)E\right(\omega \left)\right|\hfill \\ \hfill \ge & |1-ħ|\left\{2(1-j)\left|\omega \right|-\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }\left[|\ell +{\mathsf{\Lambda }}_{\ell }(1-j)|+|\ell -{\mathsf{\Lambda }}_{\ell }(1+j)|+2\aleph |\ell -{\mathsf{\Lambda }}_{\ell }|\right]{\mathsf{\varrho }}_{\ell }\left|{\omega }^{\ell }\right|\right\}\hfill \\ \hfill \ge & 2(1-j)\left|\omega \right|-\sum _{\ell =2}^{\infty }2{\mathsf{\Psi }}_{\ell }|\ell (\aleph +1)-{\mathsf{\Lambda }}_{\ell }(\aleph +j)|{\mathsf{\varrho }}_{\ell }\left|{\omega }^{\ell }\right|\ge 0\hfill \end{array}$$

or

$$\sum _{\ell =2}^{\infty }{\mathsf{\Psi }}_{\ell }|\ell (\aleph +1)-{\mathsf{\Lambda }}_{\ell }(\aleph +j)|{\mathsf{\varrho }}_{\ell }\le 1-j.$$

Conversely, suppose that (15) holds. Then, we must show

$$\Re \left\{\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })-\aleph e^{i\theta }\left[{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right]}{{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )}\right\}\ge j.$$

Upon choosing the values of $\omega$ on the positive real axis where $0\le \left|\omega \right|=r<1,$ the above inequality reduces to

$$\Re \left\{\frac{(1-j)-{\displaystyle \sum _{\ell =2}^{\infty }}{\mathsf{\Psi }}_{\ell }[\ell (1+\aleph e^{i\theta })-{\mathsf{\Lambda }}_{\ell }(j+\aleph e^{i\theta })]{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}{1-{\displaystyle \sum _{\ell =2}^{\infty }}{\mathsf{\Psi }}_{\ell }{\mathsf{\Lambda }}_{\ell }{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}\right\}\ge 0.$$

Since $\Re (-e^{i\theta })\ge -\left|e^{i\theta }\right|=-1,$ the above inequality reduces to

$$\Re \left\{\frac{(1-j)-{\displaystyle \sum _{\ell =2}^{\infty }}{\mathsf{\Psi }}_{\ell }[\ell (1+\aleph )-{\mathsf{\Lambda }}_{\ell }(j+\aleph ]{\mathsf{\varrho }}_{\ell }{r}^{\ell -1}}{1-{\displaystyle \sum _{\ell =2}^{\infty }}{\mathsf{\Psi }}_{\ell }{\mathsf{\Lambda }}_{\ell }{\mathsf{\varrho }}_{\ell }{r}^{\ell -1}}\right\}\ge 0.$$

Letting $r\to 1^{-},$ we have the desired conclusion. □

Corollary 1.

If $\mathsf{\Lambda }\left(\omega \right)\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ),$ then

$${\mathsf{\varrho }}_{\ell }\le \frac{1-j}{{\mathsf{\Psi }}_{\ell }|\ell (\aleph +1)-{\mathsf{\Lambda }}_{\ell }(\aleph +j)|}$$

where $\aleph \ge 0,|ħ|\le 1,ħ\ne 1,0\le j<1$ and ${\mathsf{\Lambda }}_{\ell }=1+ħ+\cdots +{ħ}^{\ell -1}.$

3. Neighborhood Result

A concept of the neighborhoods of an analytic mappings defined by Goodman [16], Ruscheweyh [17], and Venkateswarlu [18] for $\mathsf{\Lambda }\in T$ is as follows:

Definition 2.

Let $\aleph \ge 0,|ħ|\le 1,ħ\ne 1,0\le j<1,\alpha \ge 0$ and ${\mathsf{\Lambda }}_{\ell }=1+ħ+\cdots +{ħ}^{\ell -1}.$ We define the $\alpha -$neighborhood of a mapping $\mathsf{\Lambda }\in T$ and indicate by $N_{\alpha }\left(\mathsf{\Lambda }\right)$ consisting of all mappings $g\left(\omega \right)=\omega -{\displaystyle \sum _{\ell =2}^{\infty }}{b}_{\ell }{\omega }^{\ell }\in S(b_{\ell }\ge 0,\ell \in \mathbb{N})$ satisfying

$$\sum _{\ell =2}^{\infty }\frac{{\mathsf{\Psi }}_{\ell }|\ell (\aleph +1)-{\mathsf{\Lambda }}_{\ell }(\aleph +j)|}{1-j}|{\mathsf{\varrho }}_{\ell }-b_{\ell }|\le 1-\alpha .$$

Theorem 2.

Let $\mathsf{\Lambda }\left(\omega \right)\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)$ and $\Re \left(j\right)\ne 1.$ For any complex number ϵ with $\left|ϵ\right|<\alpha (\alpha \ge 0),$ if u fulfills the following condition:

$$\frac{\mathsf{\Lambda }\left(\omega \right)+ϵ\omega }{1+ϵ}\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)$$

then $N_{\alpha }\left(\mathsf{\Lambda }\right)\subset \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ).$

Proof. 

It is obvious that $\mathsf{\Lambda }\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)⟺$

$$\left|\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })-(\aleph e^{i\theta }+1+j)\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right)}{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })+(1-\aleph e^{i\theta }-j)\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right)}\right|<1,$$

$(-\pi <\theta <\pi ),$

for any complex number s with $\left|s\right|=1,$ we have

$$\frac{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })-(\aleph e^{i\theta }+1+j)\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right)}{(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })+(1-\aleph e^{i\theta }-j)\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right)}\ne s.$$

In other words, we must have

$$(1-s)(1-ħ)\omega {\left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)\right)}^{\prime }(1+\aleph e^{i\theta })-(\aleph e^{i\theta }+1+j+s(-1+\aleph e^{i\theta }+j))$$

$$\times \left({\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }\left(\omega \right)-{\mathfrak{J}}_{\nu }^{\delta }\mathsf{\Lambda }(ħ\omega )\right)\ne 0,$$

which is equivalent to

$$\omega -\sum _{\ell =2}^{\infty }\frac{{\mathsf{\Psi }}_{\ell }\left((\ell -{\mathsf{\Lambda }}_{\ell })(1+\aleph e^{i\theta }-s\aleph e^{i\theta })-s(\ell +{\mathsf{\Lambda }}_{\ell })-{\mathsf{\Lambda }}_{\ell }j(1-s)\right)}{j(s-1)-2s}{\omega }^{\ell }\ne 0.$$

However, $\mathsf{\Lambda }\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)⟺\frac{(\mathsf{\Lambda }\ast h)}{\omega }\ne 0,\omega \in U-\left\{0\right\},$ where $h\left(\omega \right)=\omega -{\displaystyle \sum _{\ell =2}^{\infty }}{c}_{\ell }{\omega }^{\ell }$ and

$$c_{\ell }=\frac{{\mathsf{\Psi }}_{\ell }\left((\ell -{\mathsf{\Lambda }}_{\ell })(1+\aleph e^{i\theta }-s\aleph e^{i\theta })-s(\ell +{\mathsf{\Lambda }}_{\ell })-{\mathsf{\Lambda }}_{\ell }j(1-s)\right)}{j(s-1)-2s}$$

then

$$|c_{\ell }|\le \frac{{\mathsf{\Psi }}_{\ell }\left|\ell (1+\aleph )-{\mathsf{\Lambda }}_{\ell }(\aleph +j)\right|}{1-j}$$

since $\frac{\mathsf{\Lambda }\left(\omega \right)+ϵ\omega }{1+ϵ}\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ);$ therefore, ${\omega }^{-1}\left(\frac{\mathsf{\Lambda }\left(\omega \right)+ϵ\omega }{1+ϵ}\ast h\left(\omega \right)\right)\ne 0,$ which is equivalent to

$$\frac{(\mathsf{\Lambda }\ast h)\left(\omega \right)}{(1+ϵ)\omega }+\frac{ϵ}{1+ϵ}\ne 0.$$

(17)

Now, suppose that $\left|\frac{(\mathsf{\Lambda }\ast h)\left(\omega \right)}{\omega }\right|<\alpha .$ Then, by (17), we must have

$$\begin{array}{cc}\hfill \left|\frac{(\mathsf{\Lambda }\ast h)\left(\omega \right)}{(1+ϵ)\omega }+\frac{ϵ}{1+ϵ}\right|& \ge \frac{\left|ϵ\right|}{|1+ϵ|}-\frac{1}{|1+ϵ|}\left|\frac{(\mathsf{\Lambda }\ast h)\left(\omega \right)}{\omega }\right|\hfill \\ \hfill & >\frac{\left|ϵ\right|-\alpha }{|1+ϵ|}\ge 0;\hfill \end{array}$$

this is a contradiction by $\left|ϵ\right|<\alpha$; however, we have $\left|\frac{(\mathsf{\Lambda }\ast h)\left(\omega \right)}{\omega }\right|\ge \alpha .$ If $g\left(\omega \right)=\omega -{\displaystyle \sum _{\ell =2}^{\infty }}{b}_{\ell }{\omega }^{\ell }\in N_{\alpha }\left(\mathsf{\Lambda }\right),$ then

$$\begin{array}{cc}\hfill \alpha -\left|\frac{(g\ast h)\left(\omega \right)}{\omega }\right|& \le \left|\frac{\left(\right(\mathsf{\Lambda }-g)\ast h)\left(\omega \right)}{\omega }\right|\le \sum _{\ell =2}^{\infty }|{\mathsf{\varrho }}_{\ell }-b_{\ell }\left|\right|c_{\ell }\left|\right|{\omega }^{\ell }|\hfill \\ \hfill & <\sum _{\ell =2}^{\infty }\frac{{\mathsf{\Psi }}_{\ell }|\ell (1+\aleph )-{\mathsf{\Lambda }}_{\ell }(\aleph +j)|}{1-j}|{\mathsf{\varrho }}_{\ell }-b_{\ell }|\le \alpha .\hfill \end{array}$$

□

4. Partial Sums

In this section, applying methods used by Silverman [19], Silvia [20] and also see ([21,22,23]), we investigate the ratio of a mapping $\mathsf{\Lambda }\in T$ to its sequence of partial sums ${\mathsf{\Lambda }}_m\left(\omega \right)=\omega +{\displaystyle \sum _{\ell =2}^m}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell }.$

Theorem 3.

If $\mathsf{\Lambda }\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ),$ then

$$\Re \left\{\frac{\mathsf{\Lambda }\left(\omega \right)}{{\mathsf{\Lambda }}_m\left(\omega \right)}\right\}\ge \left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}-1+j}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}\right)$$

(18)

where

$${\mathsf{\varrho }}_{\ell }={\mathsf{\varrho }}_{\ell }(\aleph ,j,{\mathsf{\Lambda }}_{\ell }){\mathsf{\Psi }}_{\ell }\ge \left\{\begin{array}{cc}1-j,\hfill & if\ell =2,3,\cdots ,m;\hfill \\ {\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1},\hfill & if\ell =m+1,m+2,\cdots \hfill \end{array}\right.$$

(19)

and

$${\mathsf{\varrho }}_{\ell }={\mathsf{\varrho }}_{\ell }(\aleph ,j,{\mathsf{\Lambda }}_{\ell })=\ell (1+\aleph )-{\mathsf{\Lambda }}_{\ell }(\aleph +j).$$

The result in (18) is sharp with the following given by

$$\mathsf{\Lambda }\left(\omega \right)=\omega +\frac{1-j}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{\omega }^{m+1}.$$

(20)

Proof. 

Defining the mapping $w,$ we may write

$$\begin{array}{cc}\hfill \frac{1+w\left(\omega \right)}{1-w\left(\omega \right)}=& \frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\left\{\frac{\mathsf{\Lambda }\left(\omega \right)}{{\mathsf{\Lambda }}_m\left(\omega \right)}-\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}-1+j}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}\right\}\hfill \\ \hfill =& \left\{\frac{1+{\displaystyle \sum _{\ell =2}^m}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}{1+{\displaystyle \sum _{\ell =2}^m}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}\right\}.\hfill \end{array}$$

(21)

It suffices to show that $\left|w\right(\omega \left)\right|\le 1.$ Now, from (21), we can obtain

$$w\left(\omega \right)=\frac{\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}{2+2{\displaystyle \sum _{\ell =2}^m}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}.$$

Hence, we obtain

$$\left|w\left(\omega \right)\right|\le \frac{\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\left|{\mathsf{\varrho }}_{\ell }\right|}{2-2{\displaystyle \sum _{\ell =2}^m}|{\mathsf{\varrho }}_{\ell }|-\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\left|{\mathsf{\varrho }}_{\ell }\right|}.$$

Now, $\left|w\right(\omega \left)\right|\le 1$, if

$$2\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right)\sum _{\ell =m+1}^{\infty }|{\mathsf{\varrho }}_{\ell }|\le 2-2\sum _{\ell =2}^m\left|{\mathsf{\varrho }}_{\ell }\right|,$$

or, equivalently,

$$\sum _{\ell =2}^m|{\mathsf{\varrho }}_{\ell }|+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right)\sum _{\ell =m+1}^{\infty }\left|{\mathsf{\varrho }}_{\ell }\right|\le 1.$$

From the condition (15), it is sufficient to show that

$$\sum _{\ell =2}^m|{\mathsf{\varrho }}_{\ell }|+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right)\sum _{\ell =m+1}^{\infty }|{\mathsf{\varrho }}_{\ell }|\le \sum _{\ell =2}^{\infty }\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }}{1-j}\left|{\mathsf{\varrho }}_{\ell }\right|$$

which is equivalent to

$$\sum _{\ell =2}^m\left(\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }-1+j}{1-j}\right)|{\mathsf{\varrho }}_{\ell }|+\sum _{\ell =m+1}^{\infty }\left(\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }-{\mathsf{\varrho }}_{\ell +1}{\mathsf{\Psi }}_{\ell +1}}{1-j}\right)\left|{\mathsf{\varrho }}_{\ell }\right|\ge 0.$$

(22)

To see that the mapping gives by (20), given the sharp result, we observe that, for $\omega =r{e}^{\frac{i\pi }{\ell }},$

$$\begin{array}{cc}\hfill \frac{\mathsf{\Lambda }\left(\omega \right)}{{\mathsf{\Lambda }}_m\left(\omega \right)}=& 1+\frac{1-j}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{\omega }^{\ell }\to 1-\frac{1-j}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}\hfill \\ \hfill =& \frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}-1+j}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}},\mathrm{when}r\to 1^{-}.\hfill \end{array}$$

□

Theorem 4.

If $\mathsf{\Lambda }\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)$, then

$$\Re \left\{\frac{{\mathsf{\Lambda }}_m\left(\omega \right)}{\mathsf{\Lambda }\left(\omega \right)}\right\}\ge \frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}+1-j},(\omega \in U)$$

(23)

where ${\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}\ge 1-j$ and

$${\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }\ge \left\{\begin{array}{cc}1-j,\hfill & if\ell =2,3,\cdots ,m;\hfill \\ {\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1},\hfill & if\ell =m+1,m+2,\cdots .\hfill \end{array}\right.$$

(24)

The result (23) is sharp with the mapping given by (20).

Proof. 

We write

$$\begin{array}{cc}\hfill \frac{1+w\left(\omega \right)}{1-w\left(\omega \right)}=& \frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}+1-j}{1-j}\left\{\frac{f_m\left(\omega \right)}{\mathsf{\Lambda }\left(\omega \right)}-\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}+1-j}\right\}\hfill \\ \hfill =& \left\{\frac{1+{\displaystyle \sum _{\ell =2}^m}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -2}-\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}{1+{\displaystyle \sum _{\ell =2}^{\infty }}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}\right\}\hfill \\ \hfill \Rightarrow \left|w\right(\omega \left)\right|& \le \frac{\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}+1-j}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\left|{\mathsf{\varrho }}_{\ell }\right|}{2-2{\displaystyle \sum _{\ell =2}^m}|{\mathsf{\varrho }}_{\ell }|-\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}+1-j}{1-j}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\left|{\mathsf{\varrho }}_{\ell }\right|}\le 1.\hfill \end{array}$$

This last inequality is equivalent to

$$\sum _{\ell =2}^m|{\mathsf{\varrho }}_{\ell }|+\sum _{\ell =m+1}^{\infty }\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{1-j}\right)\left|{\mathsf{\varrho }}_{\ell }\right|\le 1.$$

We are making use of (15)–(22). Finally, equality holds in (23) for the extremal function $\mathsf{\Lambda }\left(\omega \right)$ is given by (20). □

Theorem 5.

If $\mathsf{\Lambda }\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ),$ then

$$\begin{array}{cc}\hfill & \Re \left\{\frac{{\mathsf{\Lambda }}^{\prime }\left(\omega \right)}{{\mathsf{\Lambda }}_m^{\prime }\left(\omega \right)}\right\}\ge \left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}-(1-j)(m+1)}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}\right),(\omega \in U)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill and& \Re \left\{\frac{{\mathsf{\Lambda }}_m^{\prime }\left(\omega \right)}{{\mathsf{\Lambda }}^{\prime }\left(\omega \right)}\right\}\ge \left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}+(1-j)(m-1)}\right),(\omega \in U),\hfill \end{array}$$

(26)

where ${\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}\ge (m+1)(1-j)$ and

$${\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }\ge \left\{\begin{array}{cc}\ell (1-j),\hfill & if\ell =1,2,3,\cdots ,m;\hfill \\ \ell \left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{m+1}\right),\hfill & if\ell =m+1,m+2,\cdots .\hfill \end{array}\right.$$

The results are sharp with the mapping given by (20).

Proof. 

We write

$$\begin{array}{cc}\hfill \frac{1+w\left(\omega \right)}{1-w\left(\omega \right)}=& \left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right)\left\{\frac{{\mathsf{\Lambda }}^{\prime }\left(\omega \right)}{{\mathsf{\Lambda }}_m^{\prime }\left(\omega \right)}-\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}-(1-j)(m+1)}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}\right)\right\}\hfill \end{array}$$

where

$$w\left(\omega \right)=\frac{\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\ell {\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}{2+2{\displaystyle \sum _{\ell =2}^m}\ell {\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\ell {\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}.$$

Now, $\left|w\right(\omega \left)\right|\le 1⟺$

$$\sum _{\ell =2}^m\ell |{\mathsf{\varrho }}_{\ell }|+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right)\sum _{\ell =m+1}^{\infty }\ell \left|{\mathsf{\varrho }}_{\ell }\right|\le 1.$$

From the condition (15), it is sufficient to show that

$$\sum _{\ell =2}^m\ell |{\mathsf{\varrho }}_{\ell }|+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right)\sum _{\ell =m+1}^{\infty }\ell |{\mathsf{\varrho }}_{\ell }|\le \sum _{\ell =2}^{\infty }\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }}{1-j}\left|{\mathsf{\varrho }}_{\ell }\right|,$$

which is equivalent to

$$\sum _{\ell =2}^m\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }-(1-j)\ell }{1-j}|{\mathsf{\varrho }}_{\ell }|+\sum _{\ell =m+1}^{\infty }\frac{(m+1){\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }-\ell {\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\left|{\mathsf{\varrho }}_{\ell }\right|\ge 0.$$

To prove the inequality (26), define the mapping $w\left(\omega \right)$

$$\begin{array}{cc}\hfill \frac{1+w\left(\omega \right)}{1-w\left(\omega \right)}=& \left(\frac{(m+1)(1-j)+{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right)\left\{\frac{f_m^{\prime }\left(\omega \right)}{f^{\prime }\left(\omega \right)}-\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}+(m+1)(1-j)}\right)\right\}\hfill \end{array}$$

where

$$w\left(\omega \right)=\frac{-\left(1+\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\ell {\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}{2+2{\displaystyle \sum _{\ell =2}^m}\ell {\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}+\left(1-\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right){\displaystyle \sum _{\ell =m+1}^{\infty }}\ell {\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}}.$$

Now, $\left|w\right(\omega \left)\right|\le 1⟺$

$$\sum _{\ell =2}^m\ell |{\mathsf{\varrho }}_{\ell }|+\left(\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right)\sum _{\ell =m+1}^{\infty }\ell \left|{\mathsf{\varrho }}_{\ell }\right|\le 1.$$

(27)

It suffices to show that the left-hand side of (27) is bounded above by the condition

$$\sum _{\ell =2}^{\infty }\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }}{1-j}\left|{\mathsf{\varrho }}_{\ell }\right|,$$

which is equivalent to

$$\sum _{\ell =2}^m\left(\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }}{1-j}-\ell \right)|{\mathsf{\varrho }}_{\ell }|+\sum _{\ell =m+1}^{\infty }\left(\frac{{\mathsf{\varrho }}_{\ell }{\mathsf{\Psi }}_{\ell }}{1-j}-\frac{{\mathsf{\varrho }}_{m+1}{\mathsf{\Psi }}_{m+1}}{(m+1)(1-j)}\right)\ell \left|{\mathsf{\varrho }}_{\ell }\right|\ge 0.$$

□

5. Integral Means Property

Motivated by an integral means work of Silverman [1], many have discussed integral means results for various subclasses of $T.$ In that line inspired by the works of Ahuja et al. [24] and Thirupathi Reddy and Venkateswarlu [25] in the following theorem, we find integral mean inequality for the mappings in the class $\aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ).$

For analytic mappings u and v in $U,$u is said to be subordinate to v if there exists an analytic mapping w such that

$$w\left(0\right)=0,\left|w\right(\omega \left)\right|<1\mathrm{and}u\left(\omega \right)=v\left(w\right(\omega \left)\right),\omega \in U.$$

(28)

This subordination will be indicated here by

$$u\prec v,\omega \in U$$

or, conventionally, by

$$u\left(\omega \right)\prec v\left(\omega \right),\omega \in U.$$

In particular, when v is univalent in $U,$

$$u\prec v(\omega \in U)\iff u\left(0\right)=v\left(0\right)\mathrm{and}u\left(U\right)\subset v\left(U\right).$$

Lemma 3

([26]). If the mappings u and v are analytic in U with $u\prec v$, then

$$\underset{0}{\overset{2\pi }{\int }}{\left|u\left(\omega \right)\right|}^{\varkappa }d\theta \le \underset{0}{\overset{2\pi }{\int }}{\left|v\left(\omega \right)\right|}^{\varkappa }d\theta ,\varkappa >0,\omega =r{e}^{i\theta }\mathrm{and}0<r<1.$$

(29)

Now, we establish the integral means inequality for the mappings belonging to the class.

Theorem 6.

If $\mathsf{\Lambda }\in \aleph -{\widetilde{US}}_s(\nu ,\delta ,j,ħ)$ and ${\mathsf{\Lambda }}_2$ is defined by

$${\mathsf{\Lambda }}_2\left(\omega \right)=\omega -\frac{1-j}{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}{\omega }^2$$

(30)

then for $\omega =r{e}^{i\theta }$ and $0<r<1,$ we have

$$\underset{0}{\overset{2\pi }{\int }}{\left|\mathsf{\Lambda }\left(\omega \right)\right|}^{\varkappa }d\theta \le \underset{0}{\overset{2\pi }{\int }}{\left|{\mathsf{\Lambda }}_2\left(\omega \right)\right|}^{\varkappa }d\theta ,\varkappa >0.$$

(31)

Proof. 

Letting $\mathsf{\Lambda }$ of the form (6) and

$${\mathsf{\Lambda }}_2\left(\omega \right)=\omega -\frac{1-j}{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}{\omega }^2,$$

then we must show that

$$\underset{0}{\overset{2\pi }{\int }}{\left|1-\sum _{\ell =1}^{\infty }{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}\right|}^{\varkappa }d\theta \le \underset{0}{\overset{2\pi }{\int }}{\left|1-\frac{1-j}{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}\omega \right|}^{\varkappa }d\theta .$$

By Lemma 3, it suffices to show that

$$1-\sum _{\ell =1}^{\infty }{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}\prec 1-\frac{1-j}{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}\omega .$$

If we define the mapping $w\left(\omega \right)$ as follows:

$$w\left(\omega \right)=\sum _{\ell =2}^{\infty }\frac{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}{1-j}{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}.$$

(32)

From the above-mentioned equation,

$$w\left(0\right)=0.$$

(33)

Again, from (32), we have

$$\left|w\left(\omega \right)\right|\le \sum _{\ell =2}^{\infty }\frac{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}{1-j}|{\mathsf{\varrho }}_{\ell }{\left|\right|\omega |}^{\ell -1}.$$

Since $\omega =r{e}^{i\theta }$ and $0<r<1,$ and using (15), therefore, from the above inequality, we have

$$\left|w\left(\omega \right)\right|\le \sum _{\ell =2}^{\infty }\frac{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}{1-j}\left|{\mathsf{\varrho }}_{\ell }\right|\le 1.$$

(34)

From (32), we have

$$1-\sum _{\ell =2}^{\infty }\left|{\mathsf{\varrho }}_{\ell }\right|{\omega }^{\ell -1}=1-\frac{1-j}{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}w\left(\omega \right).$$

(35)

Since $w\left(\omega \right)$ is analytic in U, therefore, in view of Equations (28), (32), (33), and (35); inequality (34); and the subordination principle

$$1-\sum _{\ell =1}^{\infty }{\mathsf{\varrho }}_{\ell }{\omega }^{\ell -1}\prec 1-\frac{1-j}{{\mathsf{\Psi }}_2|2(\aleph +1)-{\mathsf{\Lambda }}_2(\aleph +j)|}w\left(\omega \right).$$

Since the mapping on both sides of the above relation is analytic in U, therefore, in view of Lemma 3 and Equation (30), we obtain assertion (31). This completes the proof of Theorem 6. □