arXiv subjects
Publications and source records attributed to S. Sivaprasad Kumar.
The concept of majorization is now well-known after the beautiful work of MacGregor, and then followed by Campbell in his sequel of papers. In this paper, we establish the sharp majorization results for the starlike and convex functions with respect to the conjugate points.
For $0\leqα\leq 1,$ let $H_α(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications of the form \begin{equation*} H_α(p(z),p(z)Θ(z)+zp'(z)Φ(z))\prec h(z)\Rightarrow p(z)\prec h(z), \end{equation*} where $Φ,\;Θ$ are analytic functions and $h$ is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence.
For $0<α\leq1$ and $λ>0,$ let \begin{equation}\label{1} G_{λ,α}=\left\{f\in\mathcal{A}:\left|\dfrac{1-α+αzf''(z)/f'(z)}{zf'(z)/f(z)}-(1-α)\right|<λ, z\in\mathbb{D}\right\}, \end{equation} the general form of Silverman class introduced by Tuneski and Irnak. For this class we derive some sufficient conditions in the form of differential inequalities. Further, we consider the class $Ω,$ given by \begin{equation}\label{omega} Ω=\left\{f\in\mathcal{A}:|zf'(z)-f(z)|<\dfrac{1}{2},\;z\in\mathbb{D}\right\}. \end{equation} For the above two classes, we establish inclusion relations involving some other well known subclasses of $\mathcal{S}^*$ and find radius estimates for different pairs involving these classes.
We introduce and study the following special type of differential subordination implication: \begin{equation}\label{abs} p(z)Q(z)+\frac{zp'(z)}{βp(z)+α}\prec h(z)\quad\Rightarrow p(z)\prec h(z), \end{equation} which generalizes the Briot-Bouquet differential subordination, where $Q(z)$ is analytic and $0\neqβ,α\in\mathbb{C}.$ Further, some special cases of our result are also discussed. Finally, analogues of open door lemma and integral existence theorem with applications to univalent functions are obtained.
In this paper, we generalize the Bohr-Rogosinski sum for the Ma-Minda classes of starlike and convex functions. Also the phenomenon is studied for the classes of starlike functions with respect to symmetric points and conjugate points along with their convex cases. Further, the connection between the derived results and the known ones are established with the suitable examples.
Following the trend of coefficient bound problems in Geometric Function Theory, in the present paper, we obtain the sharp bound of $|H_3(1)|$ for the class $\mathcal{S}^*$, of starlike functions and $\mathcal{SL}_q^*$, of $q$- starlike functions related with lemniscate of Bernoulli. Bound on the initial class is also an improvement over the existing known bound and the bound on the latter class generalizes the prior known outcome. Further, we determine the extremal functions to prove the sharpness of our results.
In the past several subclasses of starlike functions are defined involving real part and modulus of certain expressions of functions under study, combined by way of an inequality. In the similar fashion, we introduce a new class $\mathcal{S}^*_{q}(α)$, consisting of normalized analytic univalent functions $f$ in the open unit disk $\mathbb{D}$, satisfying $$\RE\left(\dfrac{z f'(z)}{f(z)}\right) \geq \left|1+\dfrac{z f''(z)}{f'(z)} -\dfrac{z f'(z)}{f(z)}-α\right| \quad (0 \leq α<1).$$ Evidently, $\mathcal{S}^*_{q}(α) \subset \mathcal{S}^*$, the class of starlike functions. We first establish $\mathcal{S}^*_{q}(α) \subset \mathcal{S}^*(q_α)$, the class of analytic functions $f$ satisfying $z f'(z)/f(z)\prec q_α(z),$ where $q_α$ is an extremal function. Some necessary and sufficient conditions for functions belonging to these classes are obtained in addition to the inclusion and radius problems. Further, we estimate logarithmic coefficients, inverse coefficients and Fekete-Szegö functional bounds for functions in $ \mathcal{S}^*(q_α)$.
In this paper, we prove various radius results and obtain sufficient conditions using the convolution for the Ma-Minda classes $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ of starlike and convex analytic functions. We also obtain the Bohr radius for the class $ S_{f}(ψ):= \{g(z)=\sum_{k=1}^{\infty}b_k z^k : g \prec f \}$ of subordinants, where $f\in \mathcal{S}^*(ψ).$ The results are improvements and generalizations of several well known results.
In Geometric function theory, occasionally attempts have been made to solve a particular problem for the Ma-Minda classes, $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ of univalent starlike and convex functions, respectively. Recently, a popular radius problem generally known as Bohr's phenomenon has been studied in various settings, however little is known about Rogosinski radius. In this article, for a fixed $f\in \mathcal{S}^*(ψ)$ or $\mathcal{C}(ψ),$ the class of analytic subordinants $S_{f}(ψ):= \{g : g\prec f \} $ is studied for the Bohr-Rogosinski phenomenon in a general setting. It's applications to the classes $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ are also shown.
We investigate certain properties of tilted (oblique) domains, associated with the Janowski function $(1+Az)/(1+Bz),$ where $A,B\in\mathbb{C}$ with $A\neq B$ and $|B|\leq 1$. We find several bounds for these oblique domains. Further, we extend the various known argument, radius, and subordination results involving Janowski functions with complex parameters.
Let $h$ be a non vanishing convex univalent function and $p$ be an analytic function in $\mathbb{D}$. We consider the differential subordination $$ψ_i(p(z), z p'(z)) \prec h(z)$$ with the admissible functions in consideration as $ψ_1:=(βp(z)+γ)^{-α}\left(\tfrac{(βp(z)+γ)}{β(1-α)}+ z p'(z)\right)$ and $ψ_2:=\tfrac{1}{\sqrt{γβ}}\arctan\left(\sqrt{\tfracβγ}p^{1-α}(z)\right)+\left(\tfrac{1-α}{βp^{2 (1-α)}(z)+γ}\right)\tfrac{z p'(z)}{p^α(z)}$. The objective of this paper is to find the dominants, preferably the best dominant(say $q$) of the solution of the above differential subordination satisfying $ψ_i(q, n zq'(z))= h(z)$. Further, we show that $ψ_i(q,zq'(z))= h(z)$ is an exact differential equation and $q$ is a convex univalent function in $\mathbb{D}$. In addition, we estimate the sharp lower bound of $\RE p$ for different choices of $h$ and derive a univalence criteria for functions in $\mathcal{H}$(class of analytic normalized functions) as an application to our results.
We introduce a class of analytic functions subordinate to the function $1+\sinh \left( z\right) $ and obtain various necessary and sufficient conditions for functions to be in the class. These conditions mainly comprise of the coefficient inequalities involving convolution. Further, we have obtained sharp five initial coefficients, a conjecture for the general nth coefficient and the third Hankel determinant bounds for the functions in this class. Also derived certain differential subordination implication results involving $1+\sinh \left( z\right)$.
We deal with different kinds of generalizations of $\mathcal{S}^*(ψ)$, the class of Ma-Minda starlike functions, in addition to a majorization result of $\mathcal{C}(ψ),$ the class of Ma-Minda convex functions, which are enlisted as follows: 1. Let $h$ be an analytic function, $f$ be in $\mathcal{C}(ψ)$ and $h$ be majorized by $f$ in the unit disk $\mathbb{D},$ then for a given $ψ,$ we derive a general equation, which yields the radius constant $r_ψ$ such that $|h'(z)|\leq |f'(z)|$ in $|z|\leq r_ψ$. Consequently, obtain results associating $\mathcal{S}^*(ψ)$ and others. 2. We find the largest radius $r_0$ so that the product function $g(z)h(z)/z$ belongs to a desired class for $|z|<r_0$ whenever $g\in \mathcal{S}^*(ψ_1)$ and $h\in \mathcal{S}^*(ψ_2).$ Also we obtain a condition for the functions to be in $\mathcal{S}^*(ψ)$ 3. We obtain the modified distortion theorem for $\mathcal{S}^*(ψ)$ with a general perspective. 4. For a fixed $f\in \mathcal{S}^*(ψ),$ the class of subordinants $S_{f}(ψ):= \{g : g\prec f \} $ is introduced and studied for the Bohr-phenomenon and a couple of conjectures are also proposed.
This paper deals with some radius results and inclusion relations that are established for functions in a newly defined subclass of starlike functions associated with a petal shaped domain.
We derive general formula for the fourth coefficient of the functions belonging to the Carathéodory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for certain close-to-convex functions satisfying any one of the inequalities: $\RE((1-z)f'(z))>0,$ $\RE((1-z^2)f'(z))>0,$ $\RE((1-z+z^2)f'(z))>0$ and $\RE((1-z)^2f'(z))>0$.
We introduce a general class of sense-preserving harmonic mappings defined as follows: \begin{equation*} \mathcal{S}^0_{h+\bar{g}}(M):= \{f=h+\bar{g}: \sum_{m=2}^{\infty}(γ_m|a_m|+δ_m|b_m|)\leq M, \; M>0 \}, \end{equation*} where $h(z)=z+\sum_{m=2}^{\infty}a_mz^m$, $g(z)=\sum_{m=2}^{\infty}b_m z^m$ are analytic functions in $\mathbb{D}:=\{z\in\mathbb{C}: |z|\leq1 \}$ and \begin{equation*} γ_m,\; δ_m \geq α_2:=\min \{γ_2, δ_2\}>0, \end{equation*} for all $m\geq2$. We obtain Growth Theorem, Covering Theorem and derive the Bohr radius for the class $\mathcal{S}^0_{h+\bar{g}}(M)$. As an application of our results, we obtain the Bohr radius for many classes of harmonic univalent functions and some classes of univalent functions.
We introduce the class of analytic functions $$\mathcal{F}(ψ):= \left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right) \prec ψ(z),\; ψ(0)=0 \right\},$$ where $ψ$ is univalent and establish the growth theorem with some geometric conditions on $ψ$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $α$ and $β$ for the functions in the classes $\mathcal{BS}(α):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)})-1 \prec {z}/{(1-αz^2)},\; α\in [0,1) \}$ and $\mathcal{S}_{cs}(β):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)})-1 \prec {z}/({(1-z)(1+βz)}),\; β\in [0,1) \}$, respectively which improves the earlier known bounds. The sharp Bohr-radii for the classes $S(\mathcal{BS}(α))$ and $\mathcal{BS}(α)$ are also obtained. A few examples as well as certain newly defined classes on the basis of geometry are also discussed.
In this paper, we consider the Ma-Minda classes of analytic functions $\mathcal{S}^{*}(ϕ):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)}) \prec ϕ(z) \}$ and $\mathcal{C}(ϕ):= \{f\in \mathcal{A} : (1+{zf''(z)}/{f'(z)}) \prec ϕ(z) \}$ defined on the unit disk $\mathbb{D}$ and show that the classes $\mathcal{S}^{*}(1+αz)$ and $\mathcal{C}(1+αz)$, $0<α\leq 1$ solve the problem of finding the sharp $\mathcal{S}^{*}(ϕ)$-radii and $\mathcal{C}(ϕ)$-radii for some normalized special functions, whenever $ϕ(-1)=1-α$. Radius of strongly starlikeness is also considered.