Some subordination involving polynomials induced by lower triangular matrices

Abstract

The article considers several polynomials induced by admissible lower triangular matrices and studies their subordination properties. The concept generalizes the notion of stable functions in the unit disk. Several illustrative examples, including those related to the Cesàro mean, are discussed, and connections are made with earlier works.

1 Introduction

Let $\mathcal{A}$ and $\mathcal{A}_{1}$ denote the classes consisting of functions h in the unit disk $\mathbb{D}= \{ w : |w|<1\}$ that are analytic and normalized by the conditions $\mathtt{h}(0)=0=\mathtt{h}'(0)-1$ and $\mathtt{h}(0)=1$, respectively. Further let $\mathcal{S}^{*}(\alpha _{1})$, $\alpha _{1} <1$, be the class consisting of functions $\mathtt{h} \in \mathcal{A}$ starlike of order $\alpha _{1}$ satisfying ${\mathrm{Re }} ({z \mathtt{h}'(z)}/{\mathtt{h}(z)})>\alpha _{1}$, $z \in \mathbb{D}$, and let $\mathcal{C}(\alpha _{1})$ be the class of convex functions of order $\alpha _{1}$ with the characterization ${\mathrm{Re}} ( 1+{z \mathtt{h}''(z)}/{\mathtt{h}'(z)} )> \alpha _{1}$, $z \in \mathbb{D}$. It is evident that $f \in \mathcal{C}(\alpha _{1})$ if and only if $z\mathtt{h}' \in \mathcal{S}^{*}(\alpha _{1})$. The usual classes of starlike univalent functions and convex univalent functions will be denoted respectively by $\mathcal{S}^{*}:= \mathcal{S}^{*}(0)$ and $\mathcal{C}:= \mathcal{C}(0)$. Another important subclass is $\mathcal{K}(\alpha _{1})$ consisting of $f \in \mathcal{A}$ close-to-convex of order $\alpha _{1}$ satisfying ${\mathrm{Re}} ( z\mathtt{h}'(z)/{g(z)} )>0$ for some $g \in \mathcal{S}^{\ast }(\alpha _{1})$. The class of close-to-convex univalent functions is denoted by $\mathcal{K}:= \mathcal{K}(0)$.

If f and g are analytic in $\mathbb{D}$, then f is subordinate to g, written $f(z) \prec g(z)$, if there exists an analytic self-map w in $\mathbb{D}$ satisfying $|w(z)| \leq |z|$ and $f(z)=g(w(z))$. For $f(z)=\sum_{k=0}^{\infty }a_{k} z^{k}$ and $g(z)=\sum_{k=0}^{\infty }b_{k} z^{k}$, the convolution (or Hadamard product) $f \ast g$ is given by the series $(f \ast g )(z)=\sum_{k=0}^{\infty }a_{k} b_{k} z^{k}$. The works of [4, 10, 12, 13, 18] provide good resources on these subjects.

This paper studies stable functions, a notion first introduced by Ruscheweyh and Salinas [17]. Let $\mathbb{N}$ be the set of all positive integers, and $\mathbb{N}_{0}=\mathbb{N}\cup {0}$. A function $\mathtt{h} \in \mathcal{A}_{1}$ is n-stable with respect to $F \in \mathcal{A}_{1}$ if

$$\begin{aligned} \frac{s_{n}(\mathtt{h},z)}{\mathtt{h}(z)} \prec \frac{1}{F(z)}, \end{aligned}$$

where $s_{n}(\mathtt{h},z)$ refers to the nth partial sum of the Taylor series of h about the origin. If a function h is n-stable with respect to itself, then it is known as n-stable function, while h is simply called stable (with respect to F) if h is n-stable (with respect to F) for every $n \in \mathbb{N}_{0}$.

In [11], stable functions were extended to include the $(n,\beta )$-Cesàro stable functions. By introducing an admissible lower triangular matrix, this paper aims to generalize the notion of stable functions.

Let $n \in \mathbb{N}$ be fixed and $\mathcal{G}_{n}$ be a nonempty set consisting of $(n+1) \times (n+1)$ lower triangular matrices $G=(g_{r s})$, where the entries $g_{r s} \geq 0$, $r, s=0,1, \ldots , n$, and satisfy the admissibility conditions:

• (i) $g_{r 0}=1$ for every $r=0,1,\ldots , n$,

• (ii) for each fixed $r \geq 1$, $g_{r s} = g_{r1}g_{(r-1)(s-1)}$, $s=1,\ldots , n$,

• (iii) for each fixed $r \geq 1$, $\{g_{r s}\}$ is a decreasing sequence.

The $(n+1)$th row entities in the matrix G induce a polynomial $P_{n}$ of degree n given by

$$\begin{aligned} P_{n}(z):= \sum_{k=0}^{n} g_{nk}z^{k}. \end{aligned}$$

(1)

For any $\mathtt{h}(z)= \sum_{k=0}^{\infty }a_{k} z^{k} \in \mathcal{A}_{1}$, denote by $P_{n}(\mathtt{h},z)$ the polynomial

$$\begin{aligned} P_{n}(\mathtt{h},z):= P_{n}(z) \ast \mathtt{h}(z) =\sum_{k=0}^{n} g_{nk} a_{k} z^{k}. \end{aligned}$$

(2)

It will be useful to rewrite (2) in the form

$$\begin{aligned} P_{n}(\mathtt{h},z)= g_{n1}P_{n-1}( \mathtt{h},z) + \sum_{k=0}^{n-1} ( g_{nk}- g_{n1} g_{n-1,k} ) a_{k} z^{k} + g_{nn} a_{n} z^{n}. \end{aligned}$$

(3)

Here are some examples of admissible lower triangular matrices that will be used in the sequel.

Example 1.1

Let $G=(g_{ik})$ be given by

$$ g_{i0}=1 \quad \text{and} \quad g_{ik}= \textstyle\begin{cases} 1, & 1\leq k \leq i, \\ 0, & k \geq i+1. \end{cases} $$

If $\mathtt{h}(z)= \sum_{k=0}^{\infty }a_{k} z^{k}$, $a_{0}=1$, then

$$\begin{aligned} s_{n}(\mathtt{h},z) := P_{n}(\mathtt{h},z)= P_{n}(z) \ast \mathtt{h}(z) =\sum_{k=0}^{n} a_{k} z^{k} \end{aligned}$$

is the nth partial sum of the Taylor expansion of f about the origin.

Example 1.2

For $\beta _{1} \geq 0$, let $G=(g_{ik})$ be given by

$$ g_{i0}=1 \quad \text{and} \quad g_{ik}= \textstyle\begin{cases} \frac{(1+\beta _{1})_{i-k}}{(i-k)!} \frac{i!}{(1+\beta _{1})_{i}}, & 1 \leq k \leq i, \\ 0, & k \geq i+1. \end{cases} $$

Here $(a)_{l}$ denotes the Pochhammer symbol defined by $(a)_{0}=1$ and $(a)_{l}= a(a+1)_{l-1}$, $l \in \mathbb{N}$. If $\mathtt{h}(z)= \sum_{k=0}^{\infty }a_{k} z^{k}$, $a_{0}=1$, then

$$\begin{aligned} \mathcal{S}_{n}^{\beta _{1}}(\mathtt{h},z):= P_{n}(\mathtt{h},z) = \sum_{k=0}^{n} \frac{(1+\beta _{1})_{n-k}}{(n-k)!} \frac{n!}{(1+\beta _{1})_{n}} a_{k} z^{k} \end{aligned}$$

(4)

is the nth Cesàro mean of order $\beta _{1}$ for f. When $\beta _{1}=0$, it reduces to the nth partial sum. The geometric properties of Cesàro mean can be seen in [2, 14, 15].

The admissibility condition (iii) is satisfied since, for $1 \leq k \leq i$,

$$\begin{aligned} g_{i,k-1}- g_{ik} & = \frac{(1+\beta _{1})_{i-k+1}}{(i-k+1)!} \frac{i!}{(1+\beta _{1})_{i}}- \frac{(1+\beta _{1})_{i-k}}{(i-k)!} \frac{i!}{(1+\beta _{1})_{i}} \\ & =\frac{i!}{(1+\beta _{1})_{i}} \frac{(1+\beta _{1})_{i-k}}{(i-k)!} \biggl( \frac{i-k+\beta _{1}+1}{i-k+1} - 1 \biggr)= \frac{i!}{(1+\beta _{1})_{i}}\frac{(\beta _{1})_{i-k+1}}{(i-k+1)!} \geq 0. \end{aligned}$$

Example 1.3

For $\beta \geq 0$, let $G=(g_{ik})$ be given by

$$ g_{i0}=1 \quad \text{and} \quad g_{ik}= \textstyle\begin{cases} (1-\frac{k}{i+1} )^{\beta }, & 1\leq k \leq i, \\ 0, & k \geq i+1. \end{cases} $$

If $\mathtt{h}(z)= \sum_{k=0}^{\infty }a_{k} z^{k} \in \mathcal{A}_{1}$, then

$$\begin{aligned} Q_{n}^{\beta }(\mathtt{h},z):= P_{n}( \mathtt{h},z) = \sum_{k=0}^{n} \biggl(1- \frac{k}{n+1} \biggr)^{\beta }a_{k} z^{k}. \end{aligned}$$

(5)

Now the admissibility condition (iii) holds for $1 \leq k \leq i$ because

$$\begin{aligned} g_{i,k-1}- g_{ik} & = \biggl(1-\frac{k-1}{i+1} \biggr)^{\beta }- \biggl(1- \frac{k}{i+1} \biggr)^{\beta } \\ &=\frac{(i-k+2)^{\beta }-(i-k+1)^{\beta }}{(i+1)^{\beta }} \geq 0. \end{aligned}$$

The polynomial $Q_{n}^{\beta }$ may also be expressed in terms of the Lerch transcendental function $\Phi (z, s, a)$ [1, 5, 9] given by the series

$$ \Phi (z,s,a) = \sum_{k=0}^{\infty } \frac{z^{k}}{(k+a)^{s}}, $$

(6)

with $|z|<1$, ${\mathrm{Re}} s >0$, and $a \in \mathbb{C}\backslash \{0, -1, -2, \ldots \}$. Thus

$$\begin{aligned} Q_{n}^{\beta }(z) &=\frac{(-1)^{\beta }}{(n+1)^{\beta }} \sum _{k=0}^{n} \frac{z^{k}}{(k-n-1)^{-\beta } } = \frac{(-1)^{\beta }}{(n+1)^{\beta }} \bigl( \Phi (z, -\beta , -n-1) - z^{n+1} \Phi (z, -\beta , 0) \bigr). \end{aligned}$$

Example 1.4

For $\alpha _{1} \geq 0$, let $G=(g_{ik})$ be given by

$$ g_{i0}=1 \quad \text{and} \quad g_{ik}= \textstyle\begin{cases} \frac{i+1+\delta }{i+1} \frac{i-k+1}{i-k+1+\delta }, & 1\leq k \leq i, \\ 0, & k \geq i+1. \end{cases} $$

If $f(z)= \sum_{k=0}^{\infty }a_{k} z^{k} \in \mathcal{A}_{1}$, then

$$\begin{aligned} R_{n}^{\delta }(f, z):= P_{n}(f,z) =\sum _{k=0}^{n} \biggl( \frac{n+1+\delta }{n+1} \biggr) \biggl(\frac{n-k+1}{n-k+1+\delta } \biggr) a_{k} z^{k}. \end{aligned}$$

(7)

The admissibility condition (iii) is easily seen to hold for $1 \leq k \leq i$ since

$$\begin{aligned} g_{i,k-1}- g_{ik} & = \frac{i+\delta +1}{i+1} \frac{i-k+2}{i-k+\delta +2} - \frac{i+\delta +1}{i+1} \frac{i-k+1}{i-k+\delta +1} \\ &= \frac{i+\delta +1}{i+1} \biggl(\frac{i-k+2}{i-k+\delta +2}- \frac{i-k+1}{i-k+\delta +1} \biggr) \\ &= \frac{\delta (i+\delta +1)}{(i+1)(i-k+\delta +2)(i-k+\delta +1)} \geq 0. \end{aligned}$$

In terms of the Lerch transcendental function,

$$\begin{aligned} R_{n}^{\delta } &= \frac{n+1+\delta }{n+1} \sum _{k=0}^{n} \frac{n-k+1}{n-k+1+\delta }z^{k} \\ &= \frac{n+1+\delta }{n+1} \biggl(\frac{1-z^{n+1}}{1-z} + \delta \Phi \bigl(z, 1, -(n+1+ \delta )\bigr) - \delta z^{n+1} \Phi (z, 1, - \delta ) \biggr). \end{aligned}$$

Next consider the class $\mathcal{R}^{*}(\alpha _{1})$ of prestarlike functions of order $\alpha _{1}$, $\alpha _{1} <1$, consisting of functions $f \in \mathcal{A}$ satisfying $f \ast k_{\alpha _{1}} \in \mathcal{S}^{*}(\alpha _{1})$. Here $k_{\alpha _{1}}(z) := z/(1-z)^{2-2\alpha _{1}}$ is an important member of the class $\mathcal{S}^{*}(\alpha _{1})$. Evidently, $\mathcal{R}^{*}(1/2) \equiv \mathcal{S}^{*}(1/2)$ and $\mathcal{R}^{*}(0) \equiv \mathcal{C}$.

Another two functions of importance in the sequel are $f_{\mu }$ and $V_{\lambda , \alpha _{1}}$ given by

$$\begin{aligned} f_{\mu }(z):= (1-z)^{-\mu }, \quad -1 \leq \mu \leq 1, \end{aligned}$$

(8)

and

$$\begin{aligned} \mathtt{V}_{\gamma ,\beta }(z):= \biggl(\frac{1+(1-2\beta )z}{1-z} \biggr)^{\gamma }, \quad 0 < \gamma \leq 1, 0\leq \beta \leq 1, \end{aligned}$$

(9)

for $z \in \mathbb{D}$.

A brief computation shows that

$$\begin{aligned} f_{\mu }- \frac{1-z}{\mu }f_{\mu }' &=0 \end{aligned}$$

(10)

in $z \in \mathbb{D}$.

The article [3] discusses the stability of the function $\mathtt{V}_{\gamma , \beta }$ with respect to the function $f_{\gamma }(z)=(1-z)^{-\gamma }$. It is worth to mention here that for $\beta =1/2$ the stability of $\mathtt{V}_{\gamma , 1/2}$ with respect to $f_{\gamma }$ is equivalent to the stability (with itself) of $f_{\gamma }$. Thus, the result [3, Theorem 1] is another generalization of the stability of $f_{\lambda }$ proved in [17].

Definition 1.1

(P-stable)

Let $P_{n}$ be the polynomial given by (1) which is induced by an admissible lower triangular matrix. A function $f \in \mathcal{A}_{1}$ is said to be $P_{n}$-stable with respect to $F \in \mathcal{A}_{1}$ if

$$\begin{aligned} \frac{P_{n}(f,z)}{f(z)} \prec \frac{1}{F(z)} \end{aligned}$$

holds for a fixed $n \in \mathbb{N}$. In particular, f will be called $P_{n}$-stable if it is $P_{n}$-stable with respect to itself. If f is $P_{n}$-stable (with respect to F) for every $n \in \mathbb{N}_{0}$, then f is simply said to be P-stable (with respect to F).

The article [19] also introduces a similar concept. In [19] the results encompass the generalization of the notion of the Cesàro stable functions, introduced in [11], by considering a generalized Cesàro operator

$$ \sigma _{n}^{b-1,c}(\mathtt{h},z)=\sigma _{n}^{b-1,c}(z) \ast \mathtt{h}(z), $$

where $\sigma _{n}^{b-1,c}$ is the nth Cesàro mean of order $(b-1,c)$. It is worth to mention here that $\sigma _{n}^{b-1,c}(z)$ can be induced through the admissible sequence

$$\begin{aligned} g_{i0}=1 \quad \text{and} \quad g_{ij}= \textstyle\begin{cases} \frac{B_{i-j}}{B_{i}}, & 1\leq j\leq i, \\ 0, & j \geq i+1, \end{cases}\displaystyle \end{aligned}$$

(11)

where

$$ B_{0}=1 \quad \text{and} \quad B_{i}=\frac{(b)_{i}}{(c)_{i}} \frac{1+b-c}{b} $$

with $1+b>c>0$.

The next section presents the main results involving subordination of the polynomial $P_{n}(\mathtt{V}_{\lambda , \beta },z)= P_{n}(z) \ast \mathtt{V}_{ \lambda , \beta }(z)$. The results obtained will crystallize the notion of generalized stable functions. Using the polynomials induced by the admissible matrices given in Examples 1.1–1.4, the remaining sections will be devoted to the investigations of these generalized stable functions.

2 The P-stability of $V_{\gamma ,\beta }$

The result in this section finds conditions that establish the P-stability of $\mathtt{V}_{\gamma ,\beta }$ with respect to $(1-z)^{-\gamma }$ for $1/2\leq \beta <1$ and $0< \gamma \leq 1$.

Recall a few basic properties and identities related to $\mathtt{V}_{\gamma , \beta }$ from the recent article [3]. The function $\mathtt{V}_{\gamma ,\beta }$ can be represented by the series

$$\begin{aligned} \mathtt{V}_{\gamma ,\beta }(z)=: 1+ \sum _{n=1}^{\infty }\mathtt{A}_{n}( \beta , \gamma ) z^{n}, \end{aligned}$$

(12)

with

$$\begin{aligned} \mathtt{A}_{n}:=\mathtt{A}_{n}(\beta , \gamma )=\sum_{k=0}^{n} \frac{[\gamma ]_{k}}{k!} \frac{(\gamma )_{n-k}}{(n-k)!} (1-2\beta )^{k}, \end{aligned}$$

(13)

where

$$\begin{aligned} &[y]_{k}:= y(y-1) (y-2)\cdots (y-k+1), \\ &(y)_{k}:= y(y+1) (y+2)\cdots (y+k-1), \end{aligned}$$

for $k=1, 2,\ldots$ respectively and $[y]_{0}=1=(y)_{0}$.

Also, the function $\mathtt{V}_{\gamma ,\beta }$ satisfies the differential equation

$$\begin{aligned} \mathtt{V}_{\gamma ,\beta }'(z)- \frac{\gamma (2-2\beta )}{1-2\beta z-(1-2\beta )z^{2}} \mathtt{V}_{ \gamma ,\beta }(z)=0. \end{aligned}$$

(14)

The basic properties of $\mathtt{A}_{n}$ proved in [3] are summarized in the following lemma.

Lemma 2.1

Suppose that $0 < \gamma \leq 1$and $1/2 \leq \beta \leq 1$. Then

• (a) $\mathtt{A}_{n}\geq 0$,

• (b) $(m+1)(n+1)\mathtt{A}_{n+1}-mn\mathtt{A}_{n}\geq 0$,

for all $m, n \in \mathbb{N}\cup \{0\}$.

The P-stability of $\mathtt{V}_{\gamma ,\beta }$ with respect to $(1-z)^{-\gamma }$ can be stated and proved as follows.

Theorem 2.2

Let $P_{n}$be given by (1) with $g_{i1}\leq 1$for $i\geq 1$, $\mathtt{V}_{\gamma , \beta }$by (9), and $P_{n}(f_{\mu },z)= P_{n}(z) \ast \mathtt{V}_{\gamma ,\beta }(z)$. For $0 < \gamma \leq 1$and $1/2 \leq \beta \leq 1$, the function $\mathtt{V}_{\gamma ,\beta }$is P-stable with respect to $f_{\gamma }(z)=(1-z)^{-\gamma }$.

Proof

It is enough to prove that

$$\begin{aligned} \frac{(1-z)^{\gamma } P_{n} (\mathtt{V}_{\gamma ,\beta },z)}{(1+(1-2\beta )z)^{\gamma }} \prec (1-z)^{\gamma }, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \frac{(1-z) (P_{n} (\mathtt{V}_{\gamma ,\beta },z) )^{\frac{1}{\gamma }}}{1+(1-2\beta )z} \prec (1-z). \end{aligned}$$

(15)

Define

$$\begin{aligned} &\mathtt{H}(z):=1- \frac{(1-z) (P_{n} (\mathtt{V}_{\gamma ,\beta },z) )^{\frac{1}{\gamma }}}{1+(1-2\beta )z}, \end{aligned}$$

(16)

$$\begin{aligned} &\mathtt{K}_{1}(z):= \frac{(n+1)g_{nn}A_{n+1}z^{n} -\sum_{k=0}^{n-1} (g_{n(k+1)}-g_{n k} )(k+1)\mathtt{A}_{k+1} z^{k}}{(1+(1-2\beta )z)^{2}}, \end{aligned}$$

(17)

$$\begin{aligned} &\mathtt{K}_{2}(z):= \frac{n g_{nn}A_{n}z^{n+1} +\sum_{k=2}^{n} (g_{n(k-1)}- g_{n k} )(k-1)\mathtt{A}_{k-1} z^{k}}{(1+(1-2\beta )z)^{2}}. \end{aligned}$$

(18)

Note that $|\mathtt{H}(z)|<1$ is identical with the subordination given in (15), and to prove $|\mathtt{H}(z)|<1$, it is required to show that the Taylor series of $\mathtt{H}'$ about the origin has nonnegative coefficients.

A step by step computation yields the following identities:

$$\begin{aligned} \textstyle\begin{cases} P'_{n} (\mathtt{V}_{\gamma ,\beta },z)=P_{n} (\mathtt{V}'_{\gamma , \beta },z)- (n+1)g_{nn}A_{n+1}z^{n} \\ \hphantom{P'_{n} (\mathtt{V}_{\gamma ,\beta },z)=} {}+\sum_{k=0}^{n-1} (g_{n(k+1)}-g_{n k} )(k+1) \mathtt{A}_{k+1} z^{k}, \\ z^{2} P'_{n} (\mathtt{V}_{\gamma ,\beta },z)=P_{n} (z \mathtt{V}'_{ \gamma ,\beta },z), \\ P'_{n} (z^{2}\mathtt{V}_{\gamma ,\beta },z)=P_{n} (z^{2} \mathtt{V}'_{ \gamma ,\beta },z)+ n g_{nn}A_{n}z^{n+1} \\ \hphantom{P'_{n} (z^{2}\mathtt{V}_{\gamma ,\beta },z)=} {}+\sum_{k=2}^{n} (g_{n(k-1)}-g_{n k} )(k-1) \mathtt{A}_{k-1} z^{k}. \end{cases}\displaystyle \end{aligned}$$

(19)

Differentiation of (9) along with (18) gives

$$\begin{aligned} \mathtt{H}'(z)& = \frac{(2-2\beta )P_{n} (\mathtt{V}_{\gamma ,\beta },z)}{(1+(1-2\beta )z)^{2}} \biggl( P_{n} ( \mathtt{V}_{\gamma ,\beta },z)- \frac{(1-z)(1+(1-2\beta )z)}{\gamma (2-2\beta )}P'_{n} ( \mathtt{V}_{ \gamma ,\beta },z) \biggr) \\ &= \frac{(2-2\beta )P_{n} (\mathtt{V}_{\gamma ,\beta },z)}{(1+(1-2\beta )z)^{2}} \Biggl[ P_{n} \biggl(\mathtt{V}_{\gamma ,\beta }(z)- \frac{(1-z)(1+(1-2\beta )z)}{\gamma (2-2\beta )}\mathtt{V}'_{\gamma ,\beta }(z),z \biggr) \\ &\quad {}+(n+1)g_{nn}A_{n+1}z^{n} -\sum _{k=0}^{n-1} (g_{n(k+1)}-g_{n k} ) (k+1) \mathtt{A}_{k+1} z^{k} \\ &\quad {}+(1-2\beta ) \Biggl(n g_{nn}A_{n}z^{n+1} + \sum_{k=2}^{n} (g_{n(k-1)}-g_{n k} ) (k-1)\mathtt{A}_{k-1} z^{k} \Biggr) \Biggr] \\ &=(2-2\beta )P_{n} (\mathtt{V}_{\gamma ,\beta },z) \bigl( \mathtt{K}_{1}(z)+(1-2 \beta ) \mathtt{K}_{2}(z) \bigr). \end{aligned}$$

Next we will investigate the nonnegativity of the coefficients of the Taylor series of

$$ \mathtt{K}_{1}(z)+(1-2\beta ) \mathtt{K}_{2}(z) $$

about $z=0$. A series of calculation and careful arrangement of the terms leads to

$$\begin{aligned} & \mathtt{K}_{1}(z)+(1-2\beta ) \mathtt{K}_{2}(z) \\ &\quad = \Biggl((n+1)g_{nn}A_{n+1}z^{n} -\sum _{k=0}^{n-1} (g_{n(k+1)}-g_{n k} ) (k+1) \mathtt{A}_{k+1} z^{k} \Biggr) \bigl(1+(1-2\beta )z \bigr)^{-2} \\ & \qquad {}+(1-2\beta ) \Biggl(n g_{nn}A_{n}z^{n+1} +\sum_{k=2}^{n} (g_{n(k-1)}-g_{n k} ) (k-1)\mathtt{A}_{k-1} z^{k} \Biggr) \bigl(1+(1-2\beta )z \bigr)^{-2} \\ &\quad =g_{nn} z^{n} \bigl((n+1)A_{n+1}+(1-2\beta ) n A_{n}z \bigr) \bigl(1+(1-2 \beta )z \bigr)^{-2} \\ &\qquad{}+ \Biggl( \sum_{k=0}^{n-1} (g_{nk}-g_{n (k+1)} ) (k+1) \mathtt{A}_{k+1} z^{k}+\sum_{k=2}^{n} (g_{n (k-1)}-g_{nk} ) (k-1)\mathtt{A}_{k-1} z^{k} \Biggr) \\ &\qquad{}\times \bigl(1+(1-2\beta )z \bigr)^{-2}. \end{aligned}$$

Using Lemma 2.1, it is proved in [3] that the Taylor coefficients of

$$ \bigl((n+1)A_{n+1}+(1-2\beta ) n A_{n}z \bigr) \bigl(1+(1-2 \beta )z \bigr)^{-2} $$

about $z=0$ are nonnegative. Consider the second expression

$$\begin{aligned} M&= \Biggl( \sum_{k=0}^{n-1} (g_{n(k+1)}-g_{n k} ) (k+1) \mathtt{A}_{k+1} z^{k}+(1-2\beta )\sum_{k=2}^{n} (g_{n k}-g_{n(k-1)} ) (k-1)\mathtt{A}_{k-1} z^{k} \Biggr) \\ &\quad {}\times\bigl(1+(1-2\beta )z \bigr)^{-2}. \end{aligned}$$

A computation gives

$$\begin{aligned} M &= \bigl( (g_{n 0}-g_{n 1} )\mathtt{A}_{1} \bigr) \bigl(1+(1-2 \beta )z \bigr)^{-2} \\ &\quad{}+\sum_{k=1}^{n-1} (g_{nk}-g_{n (k+1)} )z^{k} \bigl((k+1) \mathtt{A}_{k+1}+(1-2\beta )k A_{k} \bigr) \bigl(1+(1-2\beta )z \bigr)^{-2} \\ &= \bigl( (g_{n 0}-g_{n 1} )\mathtt{A}_{1} \bigr) \Biggl(\sum_{m=0}^{\infty }(m+1) (2\beta -1)^{m} z^{m} \Biggr) \\ &\quad{}+\sum_{k=1}^{n-1} \Biggl( (g_{nk}-g_{n (k+1)} )z^{k} \Biggl((k+1) A_{k+1} \\ & \quad {}+ \sum_{m=0}^{\infty } \bigl( (m+1) (k+1) \mathtt{A}_{k+1}-m k A_{k} \bigr) (2\beta -1)^{m} z^{m} \Biggr) \Biggr). \end{aligned}$$

Now by the definition of $\{g_{ij}\}$ we have $g_{nk}\geq g_{n (k+1)}$, and from Lemma 2.1, it follows that $(m+1)(k+1)\mathtt{A}_{k+1}-m k A_{k}\geq 0$. Also, $1/2 \leq \beta \leq 1$ gives $2\beta -1\geq 0$. This concludes that all the coefficients of Taylor series of $\mathtt{K}_{1}(z)+(1-2\beta ) \mathtt{K}_{2}(z)$ are nonnegative. Consequently, the coefficients of Taylor series of $\mathtt{H}'$ are nonnegative. Further, the nonnegativity of $\mathtt{A}_{n}$ implies that $|P_{n} (\mathtt{V}_{\gamma ,\beta },z)|\leq P_{n} (\mathtt{V}_{ \gamma ,\beta },|z|)$. Hence $|\mathtt{H}'(z)| \leq \mathtt{H}(|z|)$.

Since $\mathtt{H}(0)=0$ and $\mathtt{H}(1)=1$, it follows that

$$ \bigl\vert \mathtt{H}(z) \bigr\vert = \biggl\vert \int _{0}^{z} \mathtt{H}'(t) \,dt \biggr\vert \leq \biggl\vert \int _{0}^{1} \mathtt{H}'(tz) \,dt \biggr\vert \leq \int _{0}^{1} \mathtt{H}'(t) \,dt=1. $$

Therefore

$$ \biggl\vert 1- \frac{(1-z) (P_{n} (\mathtt{V}_{\gamma ,\beta },z) )^{\frac{1}{\gamma }}}{1+(1-2\beta )z} \biggr\vert < 1. $$

Equivalently,

$$ \frac{P_{n} (\mathtt{V}_{\gamma ,\beta },z)}{\mathtt{V}_{\gamma ,\beta }(z)} \prec (1-z)^{\gamma } $$

for $n \in \mathbb{N}$. Hence the conclusion. □

Choosing $\{g_{nk}\}$ as per Example 1.1, Theorem 2.2 reduces to the following result given in [3] on the stability of $\mathtt{V}_{\gamma , \beta }$.

Corollary

([3, Theorem 2.1])

For $0< \gamma \leq 1$and $1/2\leq \beta <1$, $\mathtt{V}_{\gamma , \beta }$is stable with respect to $f_{\gamma }(z)=1/(1-z)^{\gamma }$.

Consider $\{g_{nk}\}$ defined by Example 1.2, then Theorem 2.2 reduces to the following result which discusses the Cesàro stability of $\mathtt{V}_{\gamma , \beta }$.

Theorem 2.3

Suppose $\beta \geq 0$. The $0\leq \gamma \leq 1$and $1/2 \leq \beta \leq 1$, the function $\mathtt{V}_{\gamma , \beta }$given by (9) is Cesàro stable with respect to $f_{\gamma }=(1-z)^{-\gamma }$.

Consider $\{g_{nk}\}$ as given in (11), then Theorem 2.2 can be restated as follows.

Theorem 2.4

Suppose $b \geq c > 0$. The $0\leq \gamma \leq 1$and $1/2 \leq \beta \leq 1$, the function $\mathtt{V}_{\gamma , \beta }$given by (9) is generalized Cesàro stable with respect to $f_{\gamma }=(1-z)^{-\gamma }$.

For $\beta =1/2$, we have $\mathtt{V}_{\gamma , 1/2}(z)=f_{\gamma }(z)=(1-z)^{-\gamma }$ and Theorem 2.4 reduces to [19, Theorem 2.1] for $\gamma \in [0,1]$.

3 Conclusion and a future problem

The polynomials induced by the lower triangular matrices play a vital role to generalize the main concept of stable functions which was defined based on subordination and the partial sum of the power series of the function. Notably, the immediate generalization of the partial sum of power series is the Cesàro sum, which is also induced in this article through a lower triangular matrix. It is evident from the articles [6–8, 11, 16, 17, 19] that there is a strong relation between stable functions and trigonometric sums. Based on this, authors suggest investigating and developing some relevant trigonometric sums that can be induced by lower triangular matrices and can be interlinked with stable functions as future work. The extension of Theorem 2.2 for $\gamma \in [-1,0)$ is another open part that needs further investigation. Finally, the solution of the following problem may extend several ideas related to the stable functions.

Problem 3.1

Find the conditions on ϱ and γ so that

$$\begin{aligned} \frac{(1-z)^{\varrho } P_{n} (\mathtt{V}_{\gamma ,\beta },z)}{(1+(1-2\beta )z)^{\varrho }} \prec (1-z)^{\varrho }. \end{aligned}$$