Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function

G. Murugusundaramoorthy; K. Uma

Displaying similar documents to “Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function”

A note on the functions $(S_{n} f) (z)$ defined by Ruscheweyn

S. Owa, C. Y. Shen (1988)

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Some properties for $α$ -starlike functions with respect to $k$ -symmetric points of complex order

H. E. Darwish, A. Y. Lashin, S. M. Sowileh (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In the present work, we introduce the subclass $𝒯_{γ, α}^{k} (ϕ)$ , of starlike functions with respect to $k$ -symmetric points of complex order $γ$ ( $γ \neq 0$ ) in the open unit disc . Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.

Region of variability for functions with positive real part

Saminathan Ponnusamy, Allu Vasudevarao (2010)

Annales Polonici Mathematici

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For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let $_{γ, β}$ denote the class of all analytic functions P in the unit disk with P(0) = 1 and $R e (e^{i γ} P (z)) > β c o s γ$ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability $V_{} (z ₀, λ)$ for $\int_{0}^{z ₀} P (ζ) d ζ$ when P ranges over the class $(λ) = P \in_{γ, β} : P^{'} (0) = 2 (1 - β) λ e^{- i γ} c o s γ .$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

Generalized problem of starlikeness for products of close-to-star functions

Jacek Dziok (2013)

Annales Polonici Mathematici

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We consider functions of the type $F (z) = z \prod_{j = 1}^{n} {[f_{j} (z) / z]}^{a_{j}}$ , where $a_{j}$ are real numbers and $f_{j}$ are $β_{j}$ -strongly close-to-starlike functions of order $α_{j}$ . We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.

The size of the Lerch zeta-function at places symmetric with respect to the line $ℜ (s) = 1 / 2$

Ramūnas Garunkštis, Andrius Grigutis (2019)

Czechoslovak Mathematical Journal

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Let $ζ (s)$ be the Riemann zeta-function. If $t \geq 6.8$ and $σ > 1 / 2$ , then it is known that the inequality $| ζ (1 - s) | > | ζ (s) |$ is valid except at the zeros of $ζ (s)$ . Here we investigate the Lerch zeta-function $L (λ, α, s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $λ = α$ it is still possible to obtain a certain version of the inequality $| L (λ, λ, 1 - \bar{s}) | > | L (λ, λ, s) |$ .

Estimates of the real part of $\frac{f (z)}{z}$ for some classes of univalent functions

M. Obradović (1984)

Matematički Vesnik

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Convolution conditions for bounded $α$ -starlike functions of complex order

A. Y. Lashin (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f (0) = f^{^{'}} (0) - 1 = 0$ . We introduce a subclass $S_{M}^{*} (α, b)$ of $A$ , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_{M}^{*} (n, α, b)$ ( $n \geq 0$ ) related to $S_{M}^{*} (α, b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f \in A$ to belong to the class $S_{M}^{*} (α, b)$ . Several properties of the class $S_{M}^{*} (n, α, b)$ are investigated. ...

On the class of functions defined in a halfplane and starlike with respect to the boundary point

Adam Lecko (2002)

Annales Polonici Mathematici

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The purpose of this paper is to study the class $*_{\infty} (ℍ)$ of univalent analytic functions defined in the right halfplane ℍ and starlike w.r.t. the boundary point at infinity. An analytic characterization of functions in $*_{\infty} (ℍ)$ is presented.

Initial Maclaurin coefficient estimates for $λ$ -pseudo-starlike bi-univalent functions associated with Sakaguchi-type functions

Abbas Kareem Wanas, Basem Aref Frasin (2022)

Mathematica Bohemica

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We introduce and study two certain classes of holomorphic and bi-univalent functions associating $λ$ -pseudo-starlike functions with Sakaguchi-type functions. We determine upper bounds for the Taylor–Maclaurin coefficients $| a_{2} |$ and $| a_{3} |$ for functions belonging to these classes. Further we point out certain special cases for our results.

On divisibility of the class number $h^{+}$ of the real cyclotomic fields $ℚ (ζ_{p} + ζ_{p}^{- 1})$ by primes $q < 10000$

Pavel Trojovský (2000)

Mathematica Slovaca

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Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class $ℋ_{0}$ of sense-preserving harmonic functions $f = h + \bar{g}$ defined in the unit disk $| z | < 1$ and normalized so that $h (0) = 0 = h^{'} (0) - 1$ and $g (0) = 0 = g^{'} (0)$ , where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $𝒫_{H}^{0} (α)$ and $𝒢_{H}^{0} (β)$ of functions from $ℋ_{0}$ and show that if $f \in 𝒫_{H}^{0} (α)$ and $F \in 𝒢_{H}^{0} (β)$ , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $α$ and $β$ are satisfied. In the second part we study the harmonic sections...

Mean values related to the Dedekind zeta-function

Hengcai Tang, Youjun Wang (2024)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be a nonnormal cubic extension which is given by an irreducible polynomial $g (x) = x^{3} + a x^{2} + b x + c$ . Denote by $ζ_{K} (s)$ the Dedekind zeta-function of the field $K$ and $a_{K} (n)$ the number of integral ideals in $K$ with norm $n$ . In this note, by the higher integral mean values and subconvexity bound of automorphic $L$ -functions, the second and third moment of $a_{K} (n)$ is considered, i.e., $\sum_{n \leq x} a_{K}^{2} (n) = x P_{1} (log x) + O (x^{5 / 7 + ϵ}), \sum_{n \leq x} a_{K}^{3} (n) = x P_{4} (log x) + O (X^{321 / 356 + ϵ}),$ where $P_{1} (t)$ , $P_{4} (t)$ are polynomials of degree 1, 4, respectively, $ϵ > 0$ is an arbitrarily small number.

Sharp estimation of the coefficients of bounded univalent functions close to identity

Lucjan Siewierski

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CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....