• Title, Summary, Keyword: $Carath{\acute{e}}odory$ functions

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First Order Differential Subordinations for Carathéodory Functions

  • Gandhi, Shweta;Kumar, Sushil;Ravichandran, V.
    • Kyungpook Mathematical Journal
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    • v.58 no.2
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    • pp.257-270
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    • 2018
  • The well-known theory of differential subordination developed by Miller and Mocanu is applied to obtain several inclusions between $Carath{\acute{e}}odory$ functions and starlike functions. These inclusions provide sufficient conditions for normalized analytic functions to belong to certain class of Ma-Minda starlike functions.

ON SUFFICIENT CONDITIONS FOR CARATHÉODORY FUNCTIONS WITH THE FIXED SECOND COEFFICIENT

  • Kwon, Oh Sang
    • Honam Mathematical Journal
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    • v.41 no.2
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    • pp.227-242
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    • 2019
  • In the present paper, we derive several sufficient conditions for $Carath{\acute{e}}odory$ functions in the open unit disk ${\mathbb{D}}:=\{z{\in}{\mathbb{C}}:{\mid}z{\mid}<1\}$ under the constraint that the second coefficient of the function is preassigned. And, we obtain some sufficient conditions for strongly starlike functions in ${\mathbb{D}}$.

THE TILTED CARATHÉODORY CLASS AND ITS APPLICATIONS

  • Wang, Li-Mei
    • Journal of the Korean Mathematical Society
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    • v.49 no.4
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    • pp.671-686
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    • 2012
  • This paper mainly deals with the tilted Carath$\acute{e}$odory class by angle ${\lambda}$ ${\in}$ ($-{\pi}/2$, ${\pi}/2$), denoted by $P{\lambda}$) an element of which maps the unit disc into the tilted right half-plane {<${\omega}$ : Re $e^{i{\lambda}}{\omega}$ > 0}. Firstly we will characterize $P{\lambda}$ from different aspects, for example by subordination and convolution. Then various estimates of functionals over $P{\lambda}$ are deduced by considering these over the extreme points of $P{\lambda}$ or the knowledge of functional analysis. Finally some subsets of analytic functions related to $P{\lambda}$ including close-to-convex functions with argument ${\lambda}$, ${\lambda}$-spirallike functions and analytic functions whose derivative is in $P{\lambda}$ are also considered as applications.

APPLICATIONS OF DIFFERENTIAL SUBORDINATIONS TO CERTAIN CLASSES OF STARLIKE FUNCTIONS

  • Banga, Shagun;Kumar, S. Sivaprasad
    • Journal of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.331-357
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    • 2020
  • Let p be an analytic function defined on the open unit disk 𝔻. We obtain certain differential subordination implications such as ψ(p) := pλ(z)(α+βp(z)+γ/p(z)+δzp'(z)/pj(z)) ≺ h(z) (j = 1, 2) implies p ≺ q, where h is given by ψ(q) and q belongs to 𝒫, by finding the conditions on α, β, γ, δ and λ. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy |log(zf'(z)/f(z))| < 1, |(zf'(z)/f(z))2 - 1| < 1 and zf'(z)/f(z) lying in the parabolic region v2 < 2u - 1.

MULTIPLE SOLUTIONS FOR EQUATIONS OF p(x)-LAPLACE TYPE WITH NONLINEAR NEUMANN BOUNDARY CONDITION

  • Ki, Yun-Ho;Park, Kisoeb
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.6
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    • pp.1805-1821
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    • 2016
  • In this paper, we are concerned with the nonlinear elliptic equations of the p(x)-Laplace type $$\{\begin{array}{lll}-div(a(x,{\nabla}u))+{\mid}u{\mid}^{p(x)-2}u={\lambda}f(x,u) && in\;{\Omega}\\(a(x,{\nabla}u)\frac{{\partial}u}{{\partial}n}={\lambda}{\theta}g(x,u) && on\;{\partial}{\Omega},\end{array}$$ which is subject to nonlinear Neumann boundary condition. Here the function a(x, v) is of type${\mid}v{\mid}^{p(x)-2}v$ with continuous function $p:{\bar{\Omega}}{\rightarrow}(1,{\infty})$ and the functions f, g satisfy a $Carath{\acute{e}}odory$ condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.

THE SHARP BOUND OF THE THIRD HANKEL DETERMINANT FOR SOME CLASSES OF ANALYTIC FUNCTIONS

  • Kowalczyk, Bogumila;Lecko, Adam;Lecko, Millenia;Sim, Young Jae
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1859-1868
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    • 2018
  • In the present paper, we have proved the sharp inequality ${\mid}H_{3,1}(f){\mid}{\leq}4$ and ${\mid}H_{3,1}(f){\mid}{\leq}1$ for analytic functions f with $a_n:=f^{(n)}(0)/n!$, $n{\in}{\mathbb{N}},$, such that $$Re\frac{f(z)}{z}>{\alpha},\;z{\in}{\mathbb{D}}:=\{z{\in}{\mathbb{C}}:{\mid}z{\mid}<1\}$$ for ${\alpha}=0$ and ${\alpha}=1/2$, respectively, where $$H_{3,1}(f):=\left|{\array{{\alpha}_1&{\alpha}_2&{\alpha}_3\\{\alpha}_2&{\alpha}_3&{\alpha}_4\\{\alpha}_3&{\alpha}_4&{\alpha}_5}}\right|$$ is the third Hankel determinant.