• Title, Summary, Keyword: Carath$\'{e}$odory function

Search Result 13, Processing Time 0.039 seconds

First Order Differential Subordinations for Carathéodory Functions

  • Gandhi, Shweta;Kumar, Sushil;Ravichandran, V.
    • Kyungpook Mathematical Journal
    • /
    • v.58 no.2
    • /
    • pp.257-270
    • /
    • 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.

A SHARP SCHWARZ AND CARATHÉODORY INEQUALITY ON THE BOUNDARY

  • Ornek, Bulent Nafi
    • Communications of the Korean Mathematical Society
    • /
    • v.29 no.1
    • /
    • pp.75-81
    • /
    • 2014
  • In this paper, a boundary version of the Schwarz and Carath$\acute{e}$odory inequality are investigated. New inequalities of the Carath$\acute{e}$odory's inequality and Schwarz lemma at boundary are obtained by taking into account zeros of f(z) function which are different from zero. The sharpness of these inequalities is also proved.

A SHARP CARATHÉODORY'S INEQUALITY ON THE BOUNDARY

  • Ornek, Bulent Nafi
    • Communications of the Korean Mathematical Society
    • /
    • v.31 no.3
    • /
    • pp.533-547
    • /
    • 2016
  • In this paper, a generalized boundary version of $Carath{\acute{e}}odory^{\prime}s$ inequality for holomorphic function satisfying $f(z)= f(0)+a_pz^p+{\cdots}$, and ${\Re}f(z){\leq}A$ for ${\mid}z{\mid}$<1 is investigated. Also, we obtain sharp lower bounds on the angular derivative $f^{\prime}(c)$ at the point c with ${\Re}f(c)=A$. The sharpness of these estimates is also proved.

SOME RESULTS OF THE CARATHÉODORY'S INEQUALITY AT THE BOUNDARY

  • Ornek, Bulent Nafi
    • Communications of the Korean Mathematical Society
    • /
    • v.33 no.4
    • /
    • pp.1205-1215
    • /
    • 2018
  • In this paper, a boundary version of the $Carath{\acute{e}}odory^{\prime}s$ inequality is investigated. We shall give an estimate below ${\mid}f^{\prime}(b){\mid}$ according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and $z_1{\neq}0$. The sharpness of these estimates is also proved.

ON SUFFICIENT CONDITIONS FOR CARATHÉODORY FUNCTIONS WITH THE FIXED SECOND COEFFICIENT

  • Kwon, Oh Sang
    • Honam Mathematical Journal
    • /
    • v.41 no.2
    • /
    • pp.227-242
    • /
    • 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 ${\mathbb{D}}$.

SOLVABILITY FOR SECOND-ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS ON AN UNBOUNDED DOMAIN AT RESONANCE

  • Yang, Ai-Jun;Wang, Lisheng;Ge, Weigao
    • The Pure and Applied Mathematics
    • /
    • v.17 no.1
    • /
    • pp.39-49
    • /
    • 2010
  • This paper deals with the second-order differential equation (p(t)x'(t))' + g(t)f(t, x(t), x'(t)) = 0, a.e. in (0, $\infty$) with the boundary conditions $$x(0)={\int}^{\infty}_0g(s)x(s)ds,\;{lim}\limits_{t{\rightarrow}{\infty}}p(t)x'(t)=0,$$ where $g\;{\in}\;L^1[0,{\infty})$ with g(t) > 0 on [0, $\infty$) and ${\int}^{\infty}_0g(s)ds\;=\;1$, f is a g-Carath$\acute{e}$odory function. By applying the coincidence degree theory, the existence of at least one solution is obtained.

SOME REMARKS OF THE CARATHÉODORY'S INEQUALITY ON THE RIGHT HALF PLANE

  • Ornek, Bulent Nafi
    • Communications of the Korean Mathematical Society
    • /
    • v.35 no.1
    • /
    • pp.201-215
    • /
    • 2020
  • In this paper, a boundary version of Carathéodory's inequality on the right half plane for p-valent is investigated. Let Z(s) = 1+cp (s - 1)p +cp+1 (s - 1)p+1 +⋯ be an analytic function in the right half plane with ℜZ(s) ≤ A (A > 1) for ℜs ≥ 0. We derive inequalities for the modulus of Z(s) function, |Z'(0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis and finally, the sharpness of these inequalities is proved.

A PROPERTY OF CERTAIN ANALYTIC FUNCTIONS

  • Shigeyoshi Owa;Kang, Jin-Sook
    • Bulletin of the Korean Mathematical Society
    • /
    • v.32 no.2
    • /
    • pp.201-204
    • /
    • 1995
  • Let N be the class of functions of the form $$ (1.1) p(z) = 1 + p_1 z + p_2 z^2 + \cdots $$ which are analytic in the open unit disk $U = {z : $\mid$z$\mid$ < 1}$. If $p(z) \in N$ satisfies $Rep(z) > 0 (z \in U)$, then p(z) is called a Caratheodory function (cf. Goodman [2]).

  • PDF

APPLICATIONS OF DIFFERENTIAL SUBORDINATIONS TO CERTAIN CLASSES OF STARLIKE FUNCTIONS

  • Banga, Shagun;Kumar, S. Sivaprasad
    • Journal of the Korean Mathematical Society
    • /
    • v.57 no.2
    • /
    • pp.331-357
    • /
    • 2020
  • Let p be an analytic function defined on the open unit disk ��. 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 ��, 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.