• Title/Summary/Keyword: Analytic Inequalities

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SHARPENED FORMS OF ANALYTIC FUNCTIONS CONCERNED WITH HANKEL DETERMINANT

  • Ornek, Bulent Nafi
    • Korean Journal of Mathematics
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    • v.27 no.4
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    • pp.1027-1041
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    • 2019
  • In this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions. For new inequalities, the results of Jack's lemma and Hankel determinant were used. We will get a sharp upper bound for Hankel determinant H2(1). Also, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

MORSE INEQUALITIES FOR MANIFOLDS WITH BOUNDARY

  • Zadeh, Mostafa Esfahani
    • Journal of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.123-134
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    • 2010
  • The aim of this paper is to provide a proof for a version of the Morse inequalities for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on the J. Roe account of Witten's approach to Morse Theory.

CONVEXITY OF INTEGRAL OPERATORS GENERATED BY SOME NEW INEQUALITIES OF HYPER-BESSEL FUNCTIONS

  • Din, Muhey U.
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1163-1173
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    • 2019
  • In this article, we deduced some new inequalities related to hyper-Bessel function. By using these inequalities we will find some sufficient conditions under which certain families of integral operators are convex in the open unit disc. Some applications related to these results are also the part of our investigation.

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

  • Ornek, Bulent Nafi
    • Communications of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.201-215
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    • 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.

APPLICATIONS OF SUBORDINATION PRINCIPLE FOR ANALYTIC FUNCTIONS CONCERNED WITH ROGOSINSKI'S LEMMA

  • Aydinoglu, Selin;Ornek, Bulent Nafi
    • The Pure and Applied Mathematics
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    • v.27 no.4
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    • pp.157-169
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    • 2020
  • In this paper, we improve a new boundary Schwarz lemma, for analytic functions in the unit disk. For new inequalities, the results of Rogosinski's lemma, Subordinate principle and Jack's lemma were used. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

EXTREMAL LENGTH AND GEOMETRIC INEQUALITIES

  • Chung, Bohyun
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.2
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    • pp.147-156
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    • 2007
  • We introduce the extremal length and examine its properties. And we consider the geometric applications of extremal length to the boundary behavior of analytic functions, conformal mappings. We derive the theorem in connection with the capacity. This theorem applies the extremal length to the analytic function defined on the domain with a number of holes. And we obtain the theorems in connection with the pure geometric problems.

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SOME RESULTS ASSOCIATED WITH CERTAIN ANALYTIC AND UNIVALENT FUNCTIONS INVOLVING FRACTIONAL DERIVATIVE OPERATORS

  • Irmak, H.;Raina, R.K.
    • East Asian mathematical journal
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    • v.21 no.2
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    • pp.219-231
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    • 2005
  • This paper investigates some results (Theorems 2.1-2.3, below) concerning certain classes of analytic and univalent functions, involving the familiar fractional derivative operators. We state interesting consequences arising from the main results by mentioning the cases connected with the starlikeness, convexity, close-to-convexity and quasi-convexity of geometric function theory. Relevant connections with known results are also emphasized briefly.

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SUBCLASSES OF k-UNIFORMLY CONVEX AND k-STARLIKE FUNCTIONS DEFINED BY SĂLĂGEAN OPERATOR

  • Seker, Bilal;Acu, Mugur;Eker, Sevtap Sumer
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.169-182
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    • 2011
  • The main object of this paper is to introduce and investigate new subclasses of normalized analytic functions in the open unit disc $\mathbb{U}$, which generalize the familiar class of k-starlike functions. The various properties and characteristics for functions belonging to these classes derived here include (for example) coefficient inequalities, distortion theorems involving fractional calculus, extreme points, integral operators and integral means inequalities.

THE THEORY AND APPLICATIONS OF SECOND-ORDER DIFFERENTIAL SUBORDINATIONS

  • Lee, Jun Rak
    • Korean Journal of Mathematics
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    • v.7 no.1
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    • pp.85-101
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    • 1999
  • Let $p$ be analytic in the unit disc U and let $q$ be univalent in U. In addition, let ${\Omega}$ be a set in C and let ${\psi}:c^3{\times}U{\rightarrow}C$. The author determines conditions on ${\psi}$ so that $$\{{\psi}(p(z),zp^{\prime}(z),z^2p^{{\prime}{\prime}}(z);z){\mid}z{\in}U\}{\subset}{\Omega}{\Rightarrow}p(U){\subset}q(U)$$. Applications of this result to differential inequalities, differential subordinations and integral inequalities are presented.

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