A Class of Meromorphic Multivalent Functions with Negative Coefficients Defined by a Ruscheweyh-Type Operator

Abstract

We introduce and systematically study a new class $k^{\lambda ,p}(\alpha ,\beta )$ of meromorphic p-valent functions defined by means of the Ruscheweyh-type operator $D_{*}^{\lambda ,p}$, where $p\in \mathbb{N}$, $\lambda >-p$, $0\le \alpha <1$, and $\beta >0$. Membership in this class is characterized through coefficient estimates. Also investigated are growth, distortion, stability under convex combinations, radii of starlikeness and convexity of order $\rho$$(0\le \rho <1)$, convolution, the action of an integral operator of Bernardi–Libera–Livingston type, and neighborhoods.

1. Introduction

The origins of the geometric theory of functions as a branch of complex analysis concerned with the geometric properties of analytic functions can be traced back to 1851, with the foundational Riemann mapping theorem. Key milestones in its development include Koebe’s 1/4-theorem (1907), the Gronwall area theorem (1914), and de Branges’ 1984 proof of the Milin conjecture (1971), which in turn settled the Robertson conjecture (1936) and, consequently, the celebrated Bieberbach conjecture (1916). The field soon expanded to encompass new classes and subclasses of univalent and multivalent functions [1,2], both analytic and meromorphic. The volume edited by Srivastava and Owa [3] presents a comprehensive overview of the state of the discipline as of the early 1990s, and demonstrates that the introduction and systematic study of novel classes of meromorphic multivalent functions already represented a fundamental pursuit.

This body of work also showcases the analytical tools and techniques central to the field, such as convolution, differential and integral operators, coefficient estimates, and growth and distortion theorems. Extremal functions, which maximize or minimize the bounds imposed by these theorems, play a pivotal role in variational problems, where optimal function representations within a given class under specific constraints are sought. The radii of starlikeness and convexity, which determine the largest disk where a function remains starlike or convex of a given order, have direct implications for conformal mapping, clarifying how a function distorts geometric shapes. These concepts are also essential in quasiconformal analysis for controlling boundary behavior in complex dynamical systems, particularly iterated function systems [4]. Additionally, neighborhood theorems are significant in approximation theory, ensuring stability under perturbations and providing the foundation for robust numerical methods in rational approximation [5].

Although primarily theoretical, the study of these function classes finds applications across science and engineering [6,7,8]. In fluid dynamics, multivalent functions model flows around multiple obstacles, such as airfoils or porous media. In signal processing, meromorphic functions underpin analytic signal representations, essential to Fourier analysis and wavelet theory. In electrostatics and potential theory, their coefficient conditions describe charge distributions or gravitational potentials in two-dimensional domains. Thus, the investigation of meromorphic multivalent functions serves as a bridge between abstract mathematical theory and applied disciplines.

Given $p\in \mathbb{N}$, let ${\mathsf{\Sigma }}_p^{-}$ denote the class of all meromorphic functions f of the form

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{a}_n{z}^n(a_n\ge 0,n\in \mathbb{N},n\ge p;0<\left|z\right|<1)$$

(1)

which are analytic and p-valent in the punctured open unit disk of the complex plane.

The Hadamard convolution product of $g\left(z\right)={\sum }_{n=-\infty }^{\infty }{b}_n{z}^n$ and $h\left(z\right)={\sum }_{n=-\infty }^{\infty }{c}_n{z}^n$ is the formal power series

$$(g\ast h)\left(z\right)=\sum _{n=-\infty }^{\infty }{b}_n{c}_n{z}^n(0<|z|<1).$$

For $p\in \mathbb{N}$ and $\lambda >-p$, the linear operator of Ruscheweyh type $D_{*}^{\lambda ,p}:{\mathsf{\Sigma }}_p^{-}\to {\mathsf{\Sigma }}_p^{-}$ is defined as the product

$$\left(D_{*}^{\lambda ,p}f\right)\left(z\right)={\displaystyle \frac{1}{z^p{(1-z)}^{\lambda +p}}}\ast f\left(z\right)(f\in {\mathsf{\Sigma }}_p^{-}).$$

(2)

Taking into account that

$${\displaystyle \frac{1}{z^p{(1-z)}^{\lambda +p}}}={\displaystyle \frac{1}{z^p}}\sum _{n=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)z^n,$$

if $f\in {\mathsf{\Sigma }}_p^{-}$ is given by (1), the identity (2) can then be reformulated as

$$\left(D_{*}^{\lambda ,p}f\right)\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)a_n{z}^n.$$

The classical Ruscheweyh derivative was introduced by Ruscheweyh in [9] (see [10] for further context). Modified forms of this derivative have been applied to the study of multivalent analytic and meromorphic functions, as discussed, e.g., in [11] and references therein. Its variant $D_{*}^{\lambda ,p}$ appears to have been first employed to define and study new subclasses of ${\mathsf{\Sigma }}_p^{-}$ by Uralegaddi and Somanatha [12], only for $p=1$ and $\lambda \ge 0$. To our knowledge, subsequent extensions for general $p\in \mathbb{N}$ and $\lambda >-p$ remain sparse [13,14,15], a gap that the present paper aims to bridge. In contrast, the literature on the application of Ruscheweyh-type differential operators to explore novel classes of analytic or meromorphic functions with unrestricted or positive coefficients seems to be far more extensive.

Definition 1. 

Let $p\in \mathbb{N}$, $\lambda >-p$, $0\le \alpha <1$, and $\beta >0$. The class $k^{\lambda ,p}(\alpha ,\beta )$ consists of all those $f\in {\mathsf{\Sigma }}_p^{-}$, of the form (1), such that

$$\sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)[n-1+\alpha (p+1)]n{a}_n<(1-\alpha )p(p+1)$$

(3)

and

$$\left|\left(U^{\lambda ,p}f\right)\left(z\right)\right|<\beta \left|\left(V^{\lambda ,p}f\right)\left(z\right)\right|(0<|z|<1),$$

where, for $0<\left|z\right|<1$,

$$\left(U^{\lambda ,p}f\right)\left(z\right)={\displaystyle \frac{z^p{\left(D_{*}^{\lambda ,p}f\right)}^{″}\left(z\right)}{{\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)}}+(p+1)z^{p-1}$$

and

$$\left(V^{\lambda ,p}f\right)\left(z\right)={\displaystyle \frac{z^p{\left(D_{*}^{\lambda ,p}f\right)}^{″}\left(z\right)}{{\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)}}+\alpha (p+1)z^{p-1}.$$

We aim at investigating this class in a systematic manner. In the sequel, unless otherwise stated, $p, \lambda , \alpha$, and $\beta$ will denote fixed parameters constrained as in Definition 1 above. Following a standard program in geometric function theory, our study of $k^{\lambda ,p}(\alpha ,\beta )$ includes: coefficient estimates (Section 2); growth and distortion (Section 3); convex combinations (Section 4); radii of starlikeness and convexity of order $\rho$, for $0\le \rho <1$ (Section 5); convolution products (Section 6); action of an integral operator of Bernardi–Libera–Livingston type (Section 7); and neighborhoods (Section 8). The paper ends with a discussion (Section 9) on the ranges of the parameters p, $\lambda$, $\alpha$, and $\beta$ for which our results hold.

It is worth noting that, for specific parameter choices, the new class recovers known classes from the literature. Indeed, when $n\in \mathbb{N}$, the class $k^{n,1}(0,1)$ matches $T_1^{*}(0,1,-1,1,n)$ introduced in [14], which itself generalizes earlier constructions (see references therein). Similarly, for any $\alpha$ with $0\le \alpha <1$, the class $k^{0,1}(\alpha ,1)$ agrees with ${\mathsf{\Lambda }}_{1,\kappa }^{*}(\alpha ,-1,1,0)$ in [16] within a framework that, in turn, unifies several classes (again, see references therein).

2. Coefficient Estimates

As the next theorem shows, the behavior of the coefficients ensures membership in the class $k^{\lambda ,p}(\alpha ,\beta )$.

Theorem 1. 

A function $f\in {\mathsf{\Sigma }}_p^{-}$, given by (1), lies in $k^{\lambda ,p}(\alpha ,\beta )$ if, and only if,

$$\sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n{a}_n\le \beta (1-\alpha )p(p+1).$$

(4)

The estimate is sharp for any of the functions

$$\begin{array}{cc}\hfill & f_n\left(z\right)={\displaystyle \frac{1}{z^p}}\hfill \\ \hfill & -{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n}}}{z}^n(n\in \mathbb{N},n\ge p;0<|z|<1).\hfill \end{array}$$

Proof. 

Let $f\in {\mathsf{\Sigma }}_p^{-}$ be given by (1), and fix z, $0<\left|z\right|<1$. A direct computation shows that

$$\begin{array}{cc}\hfill \left|\left(U^{\lambda ,p}f\right)\left(z\right){\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)\right|& =\left|z^p{\left(D_{*}^{\lambda ,p}f\right)}^{″}\left(z\right)+(p+1)z^{p-1}{\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)\right|\hfill \\ \hfill & =\left|{\displaystyle -\sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)(p+n)n{a}_n{z}^{p+n-2}}\right|\hfill \\ \hfill & {\displaystyle <\sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)(p+n)n{a}_n}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \left|\left(V^{\lambda ,p}f\right)\left(z\right){\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)\right|\hfill \\ \hfill & =\left|z^p{\left(D_{*}^{\lambda ,p}f\right)}^{″}\left(z\right)+\alpha (p+1)z^{p-1}{\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)\right|\hfill \\ \hfill & =\left|(1-\alpha )p(p+1)z^{-2}-\sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)[n-1+\alpha (p+1)]n{a}_n{z}^{p+n-2}\right|\hfill \\ \hfill & >(1-\alpha )p(p+1)-\sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)[n-1+\alpha (p+1)]n{a}_n.\hfill \end{array}$$

If (4) holds, then

$$\begin{array}{cc}\hfill & {\displaystyle 0\le \sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)(p+n)n{a}_n}\hfill \\ \hfill & {\displaystyle \le \beta (1-\alpha )p(p+1)-\beta \sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[n-1+\alpha (p+1)\right]n{a}_n,}\hfill \end{array}$$

which implies (3). Moreover,

$$\begin{array}{cc}\hfill & \left|\left(U^{\lambda ,p}f\right)\left(z\right){\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)\right|-\beta \left|\left(V^{\lambda ,p}f\right)\left(z\right){\left(D_{*}^{\lambda ,p}f\right)}^{\prime }\left(z\right)\right|\hfill \\ \hfill & {\displaystyle <\sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n{a}_n-\beta (1-\alpha )p(p+1)\le 0.}\hfill \end{array}$$

Thus, $f\in k^{\lambda ,p}(\alpha ,\beta )$.

Conversely, assuming $f\in k^{\lambda ,p}(\alpha ,\beta )$, from

$$\left|{\displaystyle \frac{\left(U^{\lambda ,p}f\right)\left(z\right)}{\left(V^{\lambda ,p}f\right)\left(z\right)}}\right|=\left|{\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)(p+n)n{a}_n{z}^{p+n}}}{{\displaystyle (1-\alpha )p(p+1)-\sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[n-1+\alpha (p+1)\right]n{a}_n{z}^{p+n}}}}\right|<\beta ,$$

we infer

$$\Re \left[{\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)(p+n)n{a}_n{z}^{p+n}}}{{\displaystyle (1-\alpha )p(p+1)-\sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[n-1+\alpha (p+1)]n{a}_n{z}^{p+n}}}}\right]<\beta .$$

Choosing $z=r\in \mathbb{R}$ and letting $r\to 1^{-}$ yields

$${\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)(p+n)n{a}_n}}{{\displaystyle (1-\alpha )p(p+1)-\sum _{n=p}^{\infty }\left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[n-1+\alpha (p+1)]n{a}_n}}}<\beta .$$

As the denominator on the left-hand side of this inequality is positive, (4) follows.

The estimate (4) is clearly sharp for the functions $f_n$$(n\in \mathbb{N},n\ge p)$, which completes the proof. □

Corollary 1. 

If $f\in k^{\lambda ,p}(\alpha ,\beta )$ is given by (1), then

$$a_n\le {\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n}}}(n\in \mathbb{N},n\ge p).$$

This estimate is sharp for the functions $f_n$$(n\in \mathbb{N},n\ge p)$ from Theorem 1. Furthermore,

$$a_n\le {\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}(n\in \mathbb{N},n\ge p).$$

Proof. 

The first estimate is a straightforward consequence of Theorem 1. For the second, it suffices to observe that the sequence

$${\left\{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n}\right\}}_{n=p}^{\infty }$$

(5)

is increasing. □

We also have the following result.

Corollary 2. 

Suppose $f\in k^{\lambda ,p}(\alpha ,\beta )$ is given by (1). Then,

$$\sum _{n=p}^{\infty }{a}_n\le {\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}.$$

(6)

This estimate is sharp for the function $f_p$ in Theorem 1.

Proof. 

As stated above, the sequence (5) is increasing. By Theorem 1,

$$\begin{array}{cc}\hfill & \left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +2p-1}{p}}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]p\sum _{n=p}^{\infty }{a}_n\hfill \\ \hfill & \le \sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n{a}_n\hfill \\ \hfill & \le \beta (1-\alpha )p(p+1).\hfill \end{array}$$

From this, (6) follows readily. It is apparent that equality in (6) is attained for $f_p$. □

Since the sequence

$${\left\{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]}\right\}}_{n=p}^{\infty }$$

is increasing as well, we analogously deduce

Corollary 3. 

Suppose $f\in k^{\lambda ,p}(\alpha ,\beta )$ is given by (1). Then,

$$\sum _{n=p}^{\infty }n{a}_n\le {\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}.$$

(7)

This estimate is sharp for the function $f_p$ in Theorem 1.

3. Growth and Distortion

The coefficient estimates obtained in the previous section enable us to prove growth and distortion results for the class $k^{\lambda ,p}(\alpha ,\beta )$, as summarized in the following.

Theorem 2. 

If $f\in k^{\lambda ,p}(\alpha ,\beta )$, then, for $0<\left|z\right|<1$,

$$\left|f\left(z\right)\right|\ge {\displaystyle \frac{1}{{\left|z\right|}^p}}-{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^p$$

(8)

and

$$\left|f\left(z\right)\right|\le {\displaystyle \frac{1}{{\left|z\right|}^p}}+{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^p.$$

(9)

Furthermore,

$$|f^{\prime }\left(z\right)|\ge {\displaystyle \frac{p}{{\left|z\right|}^{p+1}}}-{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^{p-1}$$

(10)

and

$$|f^{\prime }\left(z\right)|\le {\displaystyle \frac{p}{{\left|z\right|}^{p+1}}}+{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^{p-1}.$$

(11)

Provided that

$$0<r<min\left\{1,{\left[{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}{\beta (1-\alpha )p(p+1)}}\right]}^{1/2p}\right\},$$

(12)

the estimates (8) and (11) are sharp for the function $f_p$ in Theorem 1 at the points $z=\pm r$, whereas the estimates (9) and (10) are sharp for the same function at the point $z=r{e}^{-i\pi /2p}$.

Proof. 

The validity of the estimates (8)–(11) for any $f\in k^{\lambda ,p}(\alpha ,\beta )$ follows immediately from Corollary 2.

Next, let r be as in (12). Then, if $z=r$, or if $z=-r$ and p is even,

$$\begin{array}{cc}\hfill |f_p\left(z\right)|& ={\displaystyle \frac{1}{r^p}}-{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{r}^p\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left|z\right|}^p}}-{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^p,\hfill \end{array}$$

while if $z=-r$ and p is odd,

$$\begin{array}{cc}\hfill |f_p\left(z\right)|& =\left|-{\displaystyle \frac{1}{r^p}}+{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{r}^p\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left|z\right|}^p}}-{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^p.\hfill \end{array}$$

This shows that (8) is sharp. Similarly, if $z=r$, or if $z=-r$ and p is odd,

$$\begin{array}{cc}\hfill |f_p^{\prime }\left(z\right)|& =\left|-{\displaystyle \frac{p}{r^{p+1}}}-{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{r}^{p-1}\right|\hfill \\ \hfill & ={\displaystyle \frac{p}{{\left|z\right|}^{p+1}}}+{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^{p-1},\hfill \end{array}$$

while if $z=-r$ and p is even,

$$\begin{array}{cc}\hfill |f_p^{\prime }\left(z\right)|& ={\displaystyle \frac{p}{r^{p+1}}}+{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{r}^{p-1}\hfill \\ \hfill & ={\displaystyle \frac{p}{{\left|z\right|}^{p+1}}}+{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^{p-1}.\hfill \end{array}$$

Therefore, (11) is also sharp. Finally, upon choosing $z=r{e}^{-i\pi /2p}$, we obtain

$$\begin{array}{cc}\hfill |f_p\left(z\right)|& =\left|i\left\{{\displaystyle \frac{1}{r^p}}+{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{r}^p\right\}\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left|z\right|}^p}}+{\displaystyle \frac{\beta (1-\alpha )(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^p,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |f_p^{\prime }\left(z\right)|& =\left|-i{e}^{i\pi /2p}\left\{{\displaystyle \frac{p}{r^{p+1}}}-{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{r}^{p-1}\right\}\right|\hfill \\ \hfill & ={\displaystyle \frac{p}{{\left|z\right|}^{p+1}}}-{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}}}{\left|z\right|}^{p-1},\hfill \end{array}$$

thus showing that (9) and (10) are sharp as well. The proof is complete. □

Remark 1. 

Concerning condition (12), see Section 9 below.

4. Stability Under Convex Combinations

Next, we gather some results about convex combinations of functions in $k^{\lambda ,p}(\alpha ,\beta )$.

Theorem 3. 

The class $k^{\lambda ,p}(\alpha ,\beta )$ is stable under convex combinations.

Proof. 

Let $f,g\in k^{\lambda ,p}(\alpha ,\beta )$, where $f\left(z\right)$ is given by (1), and g is given by

$$g\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{b}_n{z}^n(b_n\ge 0,n\in \mathbb{N},n\ge p;0<\left|z\right|<1).$$

(13)

We want to show that $tf+(1-t)g\in k^{\lambda ,p}(\alpha ,\beta )$ for all t with $0<t<1$, as the general case will follow immediately, by finite induction. In fact, fix t, $0<t<1$. We have

$$\begin{array}{cc}\hfill [tf+(1-t\left)g\right]\left(z\right)& =tf\left(z\right)+(1-t)g\left(z\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }\left[t{a}_n+(1-t)b_n\right]z^n(0<|z|<1),\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill & \sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]\left[t{a}_n+(1-t)b_n\right]n\hfill \\ \hfill & =t\sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]n{a}_n\hfill \\ \hfill & +(1-t)\sum _{n=p}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]n{b}_n\hfill \\ \hfill & \le [t+(1-t\left)\right]\beta (1-\alpha )p(p+1)=\beta (1-\alpha )p(p+1).\hfill \end{array}$$

It suffices to apply Theorem 1. □

Theorem 4. 

Define $f_0\in k^{\lambda ,p}(\alpha ,\beta )$ by

$$f_0\left(z\right)={\displaystyle \frac{1}{z^p}}(0<|z|<1),$$

and let $f_n\in k^{\lambda ,p}(\alpha ,\beta )$$(n\in \mathbb{N},n\ge p)$ be the extremal functions in Theorem 1. Then, $f\in k^{\lambda ,p}(\alpha ,\beta )$ if, and only if,

$$f\left(z\right)={\lambda }_0{f}_0+\sum _{n=p}^{\infty }{\lambda }_n{f}_n\left(z\right)(0<|z|<1)$$

(14)

for some non-negative coefficients ${\lambda }_0$, ${\lambda }_n$$(n\in \mathbb{N},n\ge p)$, such that ${\lambda }_0+{\sum }_{n=p}^{\infty }{\lambda }_n=1$.

Proof. 

Assume f is of the form (14), with ${\lambda }_0\ge 0$, ${\lambda }_n\ge 0$$(n\in \mathbb{N},n\ge p)$, and ${\lambda }_0+{\sum }_{n=p}^{\infty }{\lambda }_n=1$. For $0<\left|z\right|<1$, we then have

$$\begin{array}{cc}\hfill f\left(z\right)& =\left(1-\sum _{n=p}^{\infty }{\lambda }_n\right)f_0\left(z\right)+\sum _{n=p}^{\infty }{\lambda }_n{f}_n\left(z\right)\hfill \\ \hfill & =\left(1-\sum _{n=p}^{\infty }{\lambda }_n\right){\displaystyle \frac{1}{z^p}}\hfill \\ \hfill & +\sum _{n=p}^{\infty }{\lambda }_n\left({\displaystyle \frac{1}{z^p}}-{\displaystyle \frac{\beta (1-\alpha )p(p+1)}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n}}}{z}^n\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{\displaystyle \frac{\beta (1-\alpha )p(p+1){\lambda }_n}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n}}}{z}^n,\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill & \sum _{n=p}^{\infty }{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n}}{\beta (1-\alpha )p(p+1)}}\hfill \\ \hfill & \times {\displaystyle \frac{\beta (1-\alpha )p(p+1){\lambda }_n}{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n}}}\hfill \\ \hfill & =\sum _{n=p}^{\infty }{\lambda }_n=1-{\lambda }_0\le 1.\hfill \end{array}$$

By Theorem 1, $f\in k^{\lambda ,p}(\alpha ,\beta )$.

Conversely, assume $f\in k^{\lambda ,p}(\alpha ,\beta )$ is of the form (1), and set

$$\begin{array}{cc}\hfill {\lambda }_n& ={\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]n{a}_n}}{\beta (1-\alpha )p(p+1)}}(n\in \mathbb{N},n\ge p),\hfill \\ \hfill {\lambda }_0& =1-\sum _{n=p}^{\infty }{\lambda }_n.\hfill \end{array}$$

From Theorem 1, ${\lambda }_0$ and ${\lambda }_n$$(n\in \mathbb{N},n\ge p)$ are non-negative. Clearly, ${\lambda }_0+{\sum }_{n=p}^{\infty }{\lambda }_n=1$, and (14) holds. □

5. Starlikeness and Convexity

Let us begin by recalling the relevant definitions.

Definition 2. 

Let $0\le \rho <1$ and $0<r<1$. A function $f\in {\mathsf{\Sigma }}_p^{-}$ is said to be meromorphically multivalent starlike of order ρ in $0<\left|z\right|<r$, provided that

$$\Re \left\{-{\displaystyle \frac{1}{p}}{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right\}>\rho (0<|z|<r).$$

The radius of starlikeness of order ρ of such an f is given by

$$r_{s,\rho }\left(f\right)=sup\left\{r\in (0,1):\Re \left\{-{\displaystyle \frac{1}{p}}{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right\}>\rho (0<|z|<r)\right\}.$$

Definition 3. 

Let $0\le \rho <1$ and $0<r<1$. A function $f\in {\mathsf{\Sigma }}_p^{-}$ is said to be meromorphically multivalent convex of order ρ in $0<\left|z\right|<r$, provided that

$$\Re \left\{-{\displaystyle \frac{1}{p}}\left(1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)\right\}>\rho (0<|z|<r).$$

The radius of convexity of order ρ of such an f is given by

$$r_{c,\rho }\left(f\right)=sup\left\{r\in (0,1):\Re \left\{-{\displaystyle \frac{1}{p}}\left(1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)\right\}>\rho (0<|z|<r)\right\}.$$

The study of the classes of meromorphically multivalent starlike and meromorphically multivalent convex functions of order $\rho$, with $0\le \rho <1$, can be traced back, at least, to [17,18], respectively (see also the references therein). Since then, both classes have been the subject of a very active research, whose evolution is briefly summarized in [19]. Our next theorem determines the radii of starlikeness and convexity of order $\rho$, with $0\le \rho <1$, for the class $k^{\lambda ,p}(\alpha ,\beta )$.

Theorem 5. 

Let $f\in k^{\lambda ,p}(\alpha ,\beta )$ be given by (1), assume that the right-hand side of (6) is not greater than 1, and let $0\le \rho <1$.

(i)

The radius of starlikeness of order ρ of the function f satisfies

$$r_{s,\rho }\left(f\right)\ge \underset{n\ge p}{min}{\left\{{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]n(1-\rho )}}{\beta (1-\alpha )(p+1)[n+(2-\rho \left)p\right]}}\right\}}^{1/(p+n)}.$$

This estimate is sharp for the function $f_m$ in Theorem 1 corresponding to the index m at which the minimum is attained.

(ii)

The radius of convexity of order ρ of the function f satisfies

$$r_{c,\rho }\left(f\right)\ge \underset{n\ge p}{min}{\left\{{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]p(1-\rho )}}{\beta (1-\alpha )(p+1)[n+(2-\rho \left)p\right]}}\right\}}^{1/(p+n)}.$$

This estimate is sharp for the function $f_m$ in Theorem 1 corresponding to the index m at which the minimum is attained.

Proof. 

To prove $\left(i\right)$, it suffices to show

$$\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)+pf\left(z\right)}{pf\left(z\right)}}\right|=\left|{\displaystyle \frac{1}{p}}{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}+1\right|<1-\rho (0<|z|<r_{s,\rho }).$$

(15)

Let $0<\left|z\right|<1$. Since

$$\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)+pf\left(z\right)}{pf\left(z\right)}}\right|=\left|{\displaystyle \frac{{\displaystyle -\sum _{n=p}^{\infty }(p+n)a_n{z}^{p+n}}}{{\displaystyle p-\sum _{n=p}^{\infty }p{a}_n{z}^{p+n}}}}\right|\le {\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }(p+n)a_n{\left|z\right|}^{p+n}}}{{\displaystyle p-\sum _{n=p}^{\infty }p{a}_n{\left|z\right|}^{p+n}}}},$$

Equation (15) will follow at once from

$${\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }(p+n)a_n{\left|z\right|}^{p+n}}}{{\displaystyle p-\sum _{n=p}^{\infty }p{a}_n{\left|z\right|}^{p+n}}}}<1-\rho ,$$

or, equivalently,

$$\sum _{n=p}^{\infty }{\displaystyle \frac{n+(2-\rho )p}{p(1-\rho )}}{a}_n{\left|z\right|}^{p+n}<1.$$

By Theorem 1, for this to hold, it is enough to request that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{n+(2-\rho )p}{p(1-\rho )}}{\left|z\right|}^{p+n}\hfill \\ \hfill & <{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]n}}{\beta (1-\alpha )p(p+1)}}(n\in \mathbb{N},n\ge p),\hfill \end{array}$$

or

$$\left|z\right|<\underset{n\ge p}{inf}{\left\{{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]n(1-\rho )}}{\beta (1-\alpha )(p+1)[n+(2-\rho \left)p\right]}}\right\}}^{1/(p+n)}.$$

Similarly, to prove $\left(ii\right)$, it suffices to show

$$\left|{\displaystyle \frac{z{f}^{″}\left(z\right)+(p+1)f^{\prime }\left(z\right)}{p{f}^{\prime }\left(z\right)}}\right|=\left|{\displaystyle \frac{1}{p}}{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}+\left({\displaystyle \frac{1}{p}}+1\right)\right|<1-\rho (0<|z|<r_{c,\rho }).$$

(16)

Let $0<\left|z\right|<1$. Since the right-hand side of (7) is not greater than p, we may write

$$\left|{\displaystyle \frac{z{f}^{″}\left(z\right)+(p+1)f^{\prime }\left(z\right)}{p{f}^{\prime }\left(z\right)}}\right|=\left|{\displaystyle \frac{{\displaystyle -\sum _{n=p}^{\infty }(p+n)n{a}_n{z}^{p+n}}}{{\displaystyle -p^2-\sum _{n=p}^{\infty }pn{a}_n{z}^{p+n}}}}\right|\le {\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }(p+n)n{a}_n{\left|z\right|}^{p+n}}}{{\displaystyle p^2-\sum _{n=p}^{\infty }pn{a}_n{\left|z\right|}^{p+n}}}}.$$

Equation (16) will follow at once from

$${\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }(p+n)n{a}_n{\left|z\right|}^{p+n}}}{{\displaystyle p^2-\sum _{n=p}^{\infty }pn{a}_n{\left|z\right|}^{p+n}}}}<1-\rho ,$$

or, equivalently,

$$\sum _{n=p}^{\infty }{\displaystyle \frac{n+(2-\rho )p}{p^2(1-\rho )}}n{a}_n{\left|z\right|}^{p+n}<1.$$

By Theorem 1, for this to hold, it is enough to request that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{n+(2-\rho )p}{p^2(1-\rho )}}{\left|z\right|}^{p+n}\hfill \\ \hfill & <{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]}}{\beta (1-\alpha )p(p+1)}}(n\in \mathbb{N},n\ge p),\hfill \end{array}$$

or

$$\left|z\right|<\underset{n\ge p}{inf}{\left\{{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]p(1-\rho )}}{\beta (1-\alpha )(p+1)[n+(2-\rho \left)p\right]}}\right\}}^{1/(p+n)}.$$

Because the sequences to be minimized converge to 1 and every convergent real sequence has a smallest term, both infima are actually minima, and the assertions on the minimizing function follow immediately from the prior analysis. The proof is complete. □

Remark 2. 

Concerning the hypothesis on the size of the right-hand side of (6) in Theorem 5, see Section 9 below.

6. Convolution Products

If $f\in {\mathsf{\Sigma }}_p^{-}$ is given by (1) and

$$g\left(z\right)={\displaystyle \frac{1}{z^p}}+\sum _{n=p}^{\infty }{b}_n{z}^n(b_n\ge 0,n\in \mathbb{N},n\ge p;0<\left|z\right|<1),$$

(17)

then their Hadamard convolution $f\ast g$ takes the form

$$(f\ast g)\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{a}_n{b}_n{z}^n(0<|z|<1).$$

Theorem 6. 

Let $f\in k^{\lambda ,p}(\alpha ,\beta )$ be given by (1), and let g be given by (17). If $b_n\le 1$$(n\in \mathbb{N},n\ge p)$, then $f\ast g\in k^{\lambda ,p}(\alpha ,\beta )$.

Proof. 

By Theorem 1,

$$\sum _{n=p}^{\infty }{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n{a}_n}}{\beta (1-\alpha )p(p+1)}}\le 1.$$

Hence,

$$\begin{array}{cc}\hfill & \sum _{n=p}^{\infty }{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n{a}_n{b}_n}}{\beta (1-\alpha )p(p+1)}}\hfill \\ \hfill & \le \sum _{n=p}^{\infty }{\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)\left[p+n+\beta (n-1)+\alpha \beta (p+1)\right]n{a}_n}}{\beta (1-\alpha )p(p+1)}}\hfill \\ \hfill & \le 1.\hfill \end{array}$$

The desired conclusion derives again from Theorem 1. □

Theorem 7. 

For $f,g\in {\mathsf{\Sigma }}_p^{-}$, respectively given by (1) and (13), define $f★g\in {\mathsf{\Sigma }}_p^{-}$ to be

$$(f★g)\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{a}_n{b}_n{z}^n(0<|z|<1).$$

Assume that $f\in k^{\lambda ,p}(\alpha ,\beta )$, and either one of the following conditions holds:

(i)

$b_n\le 1$$(n\in \mathbb{N},n\ge p)$;

(ii)

$g\in k^{\lambda ,p}(\alpha ,\beta )$ and

$$\beta (1-\alpha )(p+1)\le {\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)\left[2p+\beta (p-1)+\alpha \beta (p+1)\right]}.$$

Then, $f★g\in k^{\lambda ,p}(\alpha ,\beta )$.

Proof. 

By Corollary 1, $\left(ii\right)$ implies $b_n\le 1$$(n\in \mathbb{N},n\ge p)$. And if $\left(i\right)$ holds, arguing as in the proof of Theorem 6 yields the result. □

Remark 3. 

Concerning the condition on the parameters in part $\left(ii\right)$ of Theorem 7, see Section 9 below.

7. An Integral Operator of Bernardi–Libera–Livingston Type

In this section, we investigate the action on $k^{\lambda ,p}(\alpha ,\beta )$ of the integral operator ${\mathcal{J}}_{p,c}$$(c>0)$, analog to the Bernardi–Libera–Livingston one, which has been considered in similar contexts by several authors (cf., e.g., [20,21,22] and references therein).

Theorem 8. 

Let $c>0$. If $f\in k^{\lambda ,p}(\alpha ,\beta )$, then ${\mathcal{J}}_{p,c}f\in k^{\lambda ,p}(\alpha ,\beta )$, where

$$\left({\mathcal{J}}_{p,c}f\right)\left(z\right)={\displaystyle \frac{c}{z^{p+c}}}{\int }_0^z{u}^{p+c-1}f\left(u\right)du=c{\int }_0^1{u}^{p+c-1}f\left(uz\right)du(0<|z|<1).$$

Proof. 

A direct computation shows that ${\mathcal{J}}_{p,c}f=f★g$, with

$$g\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{\displaystyle \frac{c}{p+c+n}}{z}^n(0<|z|<1).$$

Indeed, we have

$$\begin{array}{cc}\hfill \left({\mathcal{J}}_{p,c}f\right)\left(z\right)& ={\displaystyle \frac{c}{z^{p+c}}}{\int }_0^z{u}^{p+c-1}f\left(u\right)du\hfill \\ \hfill & ={\displaystyle \frac{c}{z^{p+c}}}{\int }_0^z{u}^{p+c-1}\left({\displaystyle \frac{1}{u^p}}-\sum _{n=p}^{\infty }{a}_n{u}^n\right)du\hfill \\ \hfill & ={\displaystyle \frac{c}{z^{p+c}}}\left({\int }_0^z{u}^{c-1}du-\sum _{n=p}^{\infty }{a}_n{\int }_0^z{u}^{p+c+n-1}du\right)\hfill \\ \hfill & ={\displaystyle \frac{c}{z^{p+c}}}\left({\displaystyle \frac{z^c}{c}}-\sum _{n=p}^{\infty }{a}_n{\displaystyle \frac{z^{p+c+n}}{p+c+n}}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{\displaystyle \frac{c{a}_n}{p+c+n}}{z}^n(0<|z|<1).\hfill \end{array}$$

As

$${\displaystyle \frac{c}{p+c+n}}\le 1(n\in \mathbb{N},n\ge p),$$

the desired conclusion follows from Theorem 7. □

8. Neighborhoods

After Goodman [23] and Ruscheweyh [24], neighborhoods of analytic functions have been considered by many authors. Extensions of this concept for meromorphic functions can be found, e.g., in [25,26,27]. These motivate the definitions and results in this section.

Definition 4. 

Suppose $f\in {\mathsf{\Sigma }}_p^{-}$ is given by (1), and let $0\le \delta <1$. The $N_{p,\delta }$-neighborhood of f is defined by

$$N_{p,\delta }\left(f\right)=\left\{g\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{b}_n{z}^n\in {\mathsf{\Sigma }}_p^{-}:\sum _{n=p}^{\infty }n|a_n-b_n|\le \delta \right\}.$$

Similarly, the ${\tilde{N}}_{p,\delta }$-neighborhood of f is defined by

$${\tilde{N}}_{p,\delta }\left(f\right)=\left\{g\left(z\right)={\displaystyle \frac{1}{z^p}}-\sum _{n=p}^{\infty }{b}_n{z}^n\in {\mathsf{\Sigma }}_p^{-}:\sum _{n=p}^{\infty }{\mu }_n|a_n-b_n|\le \delta \right\},$$

where

$${\mu }_n={\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +p+n-1}{n}\right)[p+n+\beta (n-1)+\alpha \beta (p+1)]n}}{\beta (1-\alpha )p(p+1)}}(n\in \mathbb{N},n\ge p).$$

Definition 5. 

Let $0\le \sigma <1$. The function $f\in {\mathsf{\Sigma }}_p^{-}$ is said to lie in the class $k_{\sigma }^{\lambda ,p}(\alpha ,\beta )$ whenever there exists $g\in k^{\lambda ,p}(\alpha ,\beta )$ such that

$$\left|{\displaystyle \frac{g\left(z\right)}{f\left(z\right)}}-1\right|\le 1-\sigma (0<|z|<1).$$

Theorem 9. 

Let $f\in k^{\lambda ,p}(\alpha ,\beta )$ be given by (1). If ${\mu }_p>1$, $0\le \delta \le 1-{\mu }_p^{-1}<1$, and

$$\sigma ={\displaystyle \frac{{\mu }_p(1-\delta )-1}{{\mu }_p-1}}\ge 0,$$

then $N_{p,\delta }\left(f\right)\subset k_{\sigma }^{\lambda ,p}(\alpha ,\beta )$.

Proof. 

Pick $g\in {\mathsf{\Sigma }}_p^{-}$ given by (13), with ${\sum }_{n=p}^{\infty }n|a_n-b_n|\le \delta$. Then, ${\sum }_{n=p}^{\infty }|a_n-b_n|\le \delta$ as well. From Corollary 2, we have ${\sum }_{n=p}^{\infty }{a}_n\le {\mu }_p^{-1}<1$. Hence,

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{g\left(z\right)}{f\left(z\right)}}-1\right|& \le {\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }|a_n-b_n{||z|}^{p+n}}}{{\displaystyle 1-\sum _{n=p}^{\infty }{a}_n{\left|z\right|}^{p+n}}}}\le {\displaystyle \frac{{\displaystyle \sum _{n=p}^{\infty }|a_n-b_n|}}{{\displaystyle 1-\sum _{n=p}^{\infty }{a}_n}}}\hfill \\ \hfill & <{\displaystyle \frac{{\mu }_p\delta }{{\mu }_p-1}}=1-\sigma (0<|z|<1),\hfill \end{array}$$

as asserted. □

Remark 4. 

Concerning the requirement that ${\mu }_p>1$ in Theorem 9, see Section 9 below.

Theorem 10. 

Assume $f\in k^{\lambda ,p}(\alpha ,\beta )$ is given by (1). If

$$\delta =1-\sum _{n=p}^{\infty }{\mu }_n{a}_n,$$

then ${\tilde{N}}_{p,\delta }\left(f\right)\subset k^{\lambda ,p}(\alpha ,\beta )$. This result is sharp, in the sense that δ cannot be increased.

Proof. 

Theorem 1 shows that

$$\sum _{n=p}^{\infty }{\mu }_n{a}_n=\xi \le 1.$$

Let $g\in {\tilde{N}}_{p,\delta }\left(f\right)$ be given by (13). We may write

$$\sum _{n=p}^{\infty }{\mu }_n{b}_n\le \sum _{n=p}^{\infty }{\mu }_n{a}_n+\sum _{n=p}^{\infty }{\mu }_n|b_n-a_n|\le \xi +\delta =1.$$

By Theorem 1, $g\in k^{\lambda ,p}(\alpha ,\beta )$.

To prove the sharpness of the result, consider the extremal function $f=f_p\in k^{\lambda ,p}(\alpha ,\beta )$ from Theorem 1:

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}-{\displaystyle \frac{1}{{\mu }_p}}{z}^p(0<|z|<1).$$

Note that, for this f, we have $\delta =0$. Choose $0<{\delta }^{*}<1$, and define $g\in {\mathsf{\Sigma }}_p^{-}$ by

$$g\left(z\right)={\displaystyle \frac{1}{z^p}}-{\displaystyle \frac{1+{\delta }^{*}}{{\mu }_p}}{z}^p(0<|z|<1).$$

Then, $g\in {\tilde{N}}_{p,{\delta }^{*}}\left(f\right)$:

$${\mu }_p\left|{\displaystyle \frac{1}{{\mu }_p}}-{\displaystyle \frac{1+{\delta }^{*}}{{\mu }_p}}\right|={\delta }^{*}.$$

However,

$${\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{n}\right)[2p+\beta (p-1)+\alpha \beta (p+1)]}}{\beta (1-\alpha )(p+1)}}{\displaystyle \frac{1+{\delta }^{*}}{{\mu }_p}}=1+{\delta }^{*}>1,$$

and Theorem 1 reveals that $g\notin k^{\lambda ,p}(\alpha ,\beta )$. □

9. Discussion

In Theorems 5, 7, and 9, the extra requirement that

$${\displaystyle \frac{{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)[2p+\beta (p-1)+\alpha \beta (p+1)]}}{\beta (1-\alpha )(p+1)}}>1$$

(18)

(equality is eventually allowed) has been imposed. It should be remarked that replacing this condition with ${\sum }_{n=p}^{\infty }{a}_n<1$ in Theorems 5 and 9 (respectively, $b_n\le 1$ for all $n\in \mathbb{N}$ in Theorem 7) actually suffices to reach the desired conclusion for a given f of the form (1) (respectively, g of the form (13)). However, (18) does not depend on the particular function chosen and, as we are about to see, is largely compatible with the other constraints on the parameters, so that the scope of the results is not only substantially preserved, but even broadened, by incorporating these assumptions from the outset.

Indeed, clearing $\beta$ in (18) yields

$$\beta >{\displaystyle \frac{{\displaystyle 2p\left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)}}{{\displaystyle (1-\alpha )(p+1)-\left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)[p-1+\alpha (p+1)]}}}$$

(19)

provided that the denominator on the right-hand side is negative, which occurs if

$${\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)[p-1+\alpha (p+1)]-(1-\alpha )(p+1)>0.}$$

(20)

Expansion of (20) reveals a positive coefficient for $\alpha$, which enables us to obtain

$$\alpha >{\displaystyle \frac{{\displaystyle p+1-\left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)(p-1)}}{(p+1)\left[{\displaystyle \left(\genfrac{}{}{0pt}{}{\lambda +2p-1}{p}\right)+1}\right]}}.$$

(21)

Thus, denoting by ${\beta }^{\lambda ,p}$ and ${\alpha }^{\lambda ,p}$ the critical values on the right-hand side of (19) and $\left(21\right)$, respectively, we find that condition (18) holds for every $p\in \mathbb{N}$, $\lambda >-p$, and $\alpha$, $\beta$ in the following ranges:

(i)

$0\le \alpha <{\alpha }^{\lambda ,p}$ and $0<\beta <{\beta }^{\lambda ,p}$;

(ii)

${\alpha }^{\lambda ,p}<\alpha <1$ and all $\beta >0$.

10. Conclusions

In this paper, we introduced and systematically investigated a new class $k^{\lambda ,p}(\alpha ,\beta )$ of meromorphic p-valent functions defined via the Ruscheweyh-type operator $D_{*}^{\lambda ,p}$$(p\in \mathbb{N},\lambda >-p,0\le \alpha <1,\beta >0)$. We characterized membership in this class through coefficient estimates. Additionally, we derived growth and distortion theorems, limiting their behavior within the unit disk. The class was shown to be stable under convex combinations. We also determined the radii of starlikeness and convexity of order $\rho$$(0\le \rho <1)$. Furthermore, we examined convolution properties and the action of an integral operator of Bernardi–Libera–Livingston type. Finally, the concept of neighborhood was explored, contributing to a deeper understanding of local behavior and approximation.

The present work suggests several avenues for future research:

• The use of a Ruscheweyh-type operator aligns with the growing trend in geometric function theory that employs differential and integral operators to define new classes of analytic and meromorphic functions. In fact, recent developments have explored generalized Ruscheweyh operators associated with fractional calculus and q-calculus, which could be incorporated into our setting for further generalizations and, hopefully, significant insights, particularly in approximation theory and combinatorics.
• Subordination remains a cornerstone technique for defining function classes and analyzing their properties. Future research on our newly introduced class might employ subordination criteria to examine extremal problems and establish sharp bounds. This direction would naturally encompass the study of higher-order coefficient estimates, including Hankel and Toeplitz determinants, as seen in contemporary research on starlike and convex functions with respect to symmetric points or other geometric constraints.
• While our work focuses on meromorphic p-valent functions, the growing interest in bi-univalent functions (those analytic and univalent in the unit disk with univalent inverses) suggests that exploring bi-p-valent analogues of our class could open new research directions.
• Our examination of neighborhoods relates directly to active research in approximation theory and partial sums. Further investigation of convergence rates and inclusion properties for functions in our class may advance geometric approximation methods, building on recent developments in the field.
• The geometric properties established in this work, particularly the distortion theorems and growth estimates, may find applications in fractional differential equations, where special functions play a fundamental role. These results could contribute to the mathematical modeling of complex physical systems that require meromorphic function techniques.
• Finally, the distortion theorems established for our class suggest promising applications beyond pure function theory. The quantitative control of geometric distortion could prove valuable in signal processing for analyzing phase distortions and in computational conformal mappings where precise control of boundary behavior is essential.

We believe that further exploration of these directions may yield additional perspectives into both the theoretical aspects and potential applications of the new class and related function families.