arXiv subjects

S. Sivaprasad Kumar

Publications and source records attributed to S. Sivaprasad Kumar.

At least 19 recordsLinked to original sources

arXiv2026

On Geometric properties and Coefficient bounds for $\mathcal{S}^*_{B}$

This paper deals with the geometric properties of functions belonging to the class $\mathcal{S}^*_{B}$ of starlike functions associated with a balloon-shaped domain, given by \[ \mathcal{S}^{\ast}_{B}= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log (1+z)} :=B(z), \quad z \in \mathbb{D} \right\}, \] and also derive sharp bounds for the Zalcman functionals, Krushkal inequality, third-order Hankel, Toeplitz and Hermitian-Toeplitz determinant. The sharpness of these results are verified by constructing suitable extremal functions.

math.CV↗

arXiv2026

Sharp Coefficient Bounds for certain $q$-Starlike Functions

Geometric function theory increasingly draws on $q$-calculus to model discrete and quantum-inspired phenomena. Motivated by this, the present paper introduces new subclasses of analytic functions: the class $\mathcal{S}^{*}_{ξ_q}$ of $q$-starlike functions associated with the Ma-Minda function $ξ_q(z)$, and its limiting classical counterpart $\mathcal{S}^{*}_ξ$ associated with $ξ(z)$, where $q \in (0,1)$. We systematically establish sharp coefficient estimates including the Fekete-Szegö, Hankel and Toeplitz determinants. We establish the sharpness of the $q$-coefficient estimates using a newly derived integral representation, which offers a more effective alternative to the conventional convolution-based extremal construction. It is further shown that all $q$-results reduce to their classical counterparts as $q \to 1^{-}$.

math.CV↗

arXiv2026

Coefficient problems of Starlike Functions Related to a Balloon-Shaped Domain

Recent advances in image and signal processing have drawn on geometric function theory, particularly coefficient estimate problems. Motivated by their significance, we introduce a class of starlike functions related to a balloon-shaped domain \[ \mathcal{S}^*_{\mathcal{B}}= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} := B(z); \; z \in \mathbb{D} \right\}, \] where $B(z)$ maps the unit disk $\mathbb{D}$ onto a balloon-shaped domain. This work establishes bounds for the second order Hankel determinants and second order Toeplitz determinants involving the initial coefficients, the logarithmic coefficients and the logarithmic coefficients of the inverse function for $f \in \mathcal{S}^*_{\mathcal{B}}$

math.CV↗

arXiv2026

Generalized Toeplitz determinants for Starlike Mappings in Several Complex Variables

This paper establishes sharp bounds for the second and third-order Toeplitz determinants associated with starlike functions $f$ in the unit disk such that $f(z)-z$ has a zero of order $k+1$ at $z=0$. These bounds are further extended to starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$. The derived results generalize several known bounds as special cases.

math.CV↗

arXiv2024

On Coefficient problems for \textbf{$S^*_ρ$}

Logarithmic and inverse logarithmic coefficients play a crucial role in the theory of univalent functions. In this study, we focus on the class of starlike functions $\mathcal{S}^*_ρ$, defined as \[ \mathcal{S}^*_ρ= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec ρ(z), \; z \in \mathbb{D} \right\}, \] where $ρ(z) := 1 + \sinh^{-1}(z)$, which maps the unit disk $\mathbb{D}$ onto a petal-shaped domain. This investigation aims to establish bounds for the second Hankel and Toeplitz determinants, with their entries determined by the logarithmic coefficients of $f$ and its inverse $f^{-1}$, for functions $f \in \mathcal{S}^*_ρ$.

math.CV↗

arXiv2024

Starlike Functions Associated with a Non-Convex Domain

We introduce and study a class of starlike functions associated with the non-convex domain \[ \mathcal{S}^*_{nc} = \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1+z}{\cos{z}} =: φ_{nc}(z), \;\; z \in \mathbb{D} \right\}. \] Key results include the growth and distortion theorems, initial coefficient bounds, and the sharp estimates for third-order Hankel and Hermitian-Toeplitz determinants. We also examine inclusion relations, radius problems for certain subclasses, and subordination results. These findings enrich the theory of starlike functions associated with non-convex domains, offering new perspectives in geometric function theory.

math.CV↗

arXiv2024

On Sufficient Conditions for the class $S^*_\cosh\sqrt{z}$

Using differential subordination technique, such as Briot-Bouquet and others, we establish sufficient conditions for functions to be in a class $\mathcal{S}^{*}_{\varrho},$ consisting of starlike functions that are associated with $\varrho(z):=\cosh \sqrt{z}.$ %Further, using admissibility conditions, some differential subordination results for $\mathcal{S}^{*}_{\varrho}$ are derived. Furthermore, by employing admissibility conditions, we obtain various differential subordination results pertaining to the class $\mathcal{S}^{*}_{\varrho}.$

math.CV↗

arXiv2024

A new differential subordination technique for a subclass of starlike functions

In the present investigation, we employ a new technique to find several first and second order differential subordination implications involving the following starlike class associated with a bean shaped domain: \begin{equation*} \mathcal{S}^*_{\mathfrak{B}}:=\left\{f\in\mathcal{S}:\dfrac{zf'(z)}{f(z)}\prec\sqrt{1+\tanh{z}}=:\mathfrak{B}(z)\right\}. \end{equation*} Also, we give several applications stemming from our derived results.

math.CV↗

arXiv2024

On Applications of Extended Jack's Lemma

We introduce and study the class ${\bf\mathcal{G}}(α,β)$ comprising analytic functions associated with a sector domain, where $α,β\in(0,1]$. Using the extended version of Jack's lemma, we deduce Open-Door lemma type sufficient conditions for functions to be in ${\bf\mathcal{G}}(α,β)$. Furthermore, we point out special cases of our results that align with known results.

math.CV↗

arXiv2024

On estimation of Hankel determinants for certain class of starlike functions

In the present study, we consider two subclasses starlike and convex functions, denoted by $\mathcal{S}_{\mathcal{B}}^{*}$ and $\mathcal{C}_{\mathcal{B}}$ respectively, associated with a bean-shaped domain. Further, we estimate certain sharp initial coefficients, as well as second, third and fourth-order Hankel determinants for functions belonging to the class $\mathcal{S}_{\mathcal{B}}^{*}$. Additionally, we compute sharp second and third-order Hankel determinants for functions belonging to the $\mathcal{C}_{\mathcal{B}}$ class.

math.CV↗

arXiv2024

Higher order differential subordinations for certain starlike functions

In this paper, we employ a novel second and third-order differential subordination technique to establish the sufficient conditions for functions to belong to the classes $\mathcal{S}^*_s$ and $\mathcal{S}^*_ρ$, where $\mathcal{S}^*_s$ is the set of all normalized analytic functions $f$ satisfying $ zf'(z)/f(z)\prec 1+\sin z$ and $\mathcal{S}^*_ρ$ is the set of all normalized analytic functions $f$ satisfying $ zf'(z)/f(z)\prec 1+\sinh^{-1} z$.

math.CV↗

arXiv2024

Second and Third order differential subordination for exponential function

This article presents several findings regarding second and third-order differential subordination of the form: $$ p(z)+γ_1 zp'(z)+γ_2 z^2p''(z)\prec h(z)\implies p(z)\prec e^z $$ and $$ p(z)+γ_1 zp'(z)+γ_2 z^2p''(z)+γ_3 z^3p'''(z)\prec h(z)\implies p(z)\prec e^z. $$ Here, $γ_1$, $γ_2$, and $γ_3$ represent positive real numbers, and various selections of $h(z)$ are explored within the context of the class $\mathcal{S}^{*}_{e} := \{f \in \mathcal{A} : zf'(z)/f(z) \prec e^z\}$, which denotes the class of starlike functions associated with the exponential function.

math.CV↗

arXiv2024

On Starlike Functions Associated with a Bean Shaped Domain

In this paper, we introduce and explore a new class of starlike functions denoted by $\mathcal{S}^*_{\mathfrak{B}}$, defined as follows: $$\mathcal{S}^*_{\mathfrak{B}}=\{f\in \mathcal{A}:zf'(z)/f(z)\prec \sqrt{1+\tanh{z}}=:\mathfrak{B}(z)\}.$$ Here, $\mathfrak{B}(z)$ represents a mapping from the unit disk onto a bean-shaped domain. Our study focuses on understanding the characteristic properties of both $\mathfrak{B}(z)$ and the functions in $\mathcal{S}^*_{\mathfrak{B}}$. We derive sharp conditions under which $ψ(p)\prec\sqrt{1+\tanh(z)}$ implies $p(z)\prec ((1+A z)/(1+B z))^γ$, where $ψ(p)$ is defined as: \begin{equation*} (1-α)p(z)+αp^2(z)+β\frac{zp'(z)}{p^k(z)}\quad \text{and}\quad (p(z))^δ+β\frac{zp'(z)}{(p(z))^k}. \end{equation*} Additionally, we establish inclusion relations involving $\mathcal{S}^*_{\mathfrak{B}}$ and derive precise estimates for the sharp radii constants of $\mathcal{S}^*_{\mathfrak{B}}$.

math.CV↗

arXiv2024

Differential Subordination of Certain Class of Starlike Functions

This paper presents several results concerning second and third-order differential subordination for the class $\mathcal{S}^{*}_{e}:=\{f\in \mathcal{A}:zf'(z)/f(z)\prec e^z\}$, which represents the class of starlike functions associated with exponential function.

math.CV↗

arXiv2023

On a Subclass of Starlike Functions Associated with a Strip Domain

In the present investigation, we introduce a new subclass of starlike functions defined by $\mathcal{S}^{*}_τ:=\{f\in \mathcal{A}:zf'(z)/f(z) \prec 1+\arctan z=:τ(z)\}$, where $τ(z)$ maps the unit disk $\mathbb {D}:= \{z\in \mathbb{C}:|z|<1\}$ onto a strip domain. We derive structural formulae, growth, and distortion theorems for $\mathcal{S}^{*}_τ$. Also, inclusion relations with some well-known subclasses of $\mathcal{S}$ are established and obtain sharp radius estimates, as well as sharp coefficient bounds for the initial five coefficients and the second and third-order Hankel determinants of $\mathcal{S}^{*}_τ$.

math.CV↗

arXiv2023

Sharp Estimate of Fifth Coefficient for Ma Minda Starlike and Convex Functions

\noindent In the present investigation, we find the sharp bound of fifth coefficient of analytic normalized function $f$ satisfying $z f'(z)/f(z) \prec φ(z)$ when coefficients of $φ$ satisfy certain conditions. For an appropriate choice of $φ$, the already known estimates for various other subclasses of starlike functions follow directly from the obtained result.

math.CV↗

arXiv2023

Certain Coefficient Problems of $\mathcal{S}_{e}^{*}$ and $\mathcal{C}_{e}$

In this current study, we consider the classes $\mathcal{S}^{*}_{e}$ and $\mathcal{C}_e$ to obtain sharp bounds for the third Hankel determinant for functions within these classes. Additionally, we provide estimates for the sixth and seventh coefficients while establishing the fourth-order Hankel determinant as well.

math.CV↗

arXiv2023

On sharp third Hankel determinant for certain starlike functions

In this paper, we provide an estimation for the sharp bound of the third Hankel determinant of starlike functions of order $α$, where $α$ ranges in the interval $[0, 1/6]\cup \{1/2\}$ and thereby extending the result of Rath et al. (Complex Anal Oper Theory: No. 65, 16(5), 8 pp 2022).

math.CV↗