Now showing 1 - 10 of 225
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    Convexity of integral transforms and function spaces
    (01-01-2007)
    Balasubramanian, R.
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    Prabhakaran, D. J.
    For β<1, let Pγ(β) denote the class of all normalized analytic functions f in the unit disc Δ such that for some φ ε ℝ. Let script S sign*(μ), 0 ≤ μ<1, denote the usual class of starlike functions of order μ. Define script K sign (μ)={f: zf′(z) ε script S sign *(μ)}, the class of all convex functions of order μ. In this paper, we consider integral transforms of the form The aim of this paper is to find conditions on λ (t) so that each of the transformations carries script P signγ(β) into script S sign *(μ) or script K sign(μ). A number of applications for certain special choices of λ (t) are also established. These results extend the previously known results by a number of authors.
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    Improved Bohr's phenomenon in quasi-subordination classes
    (01-02-2022) ;
    Vijayakumar, Ramakrishnan
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    Wirths, Karl Joachim
    Recently the present authors established refined versions of Bohr's inequality in the case of bounded analytic functions. In this article, we state and prove a generalization of these results. Here, we consider the image of the origin and the boundary of the image of the unit disk under the function in question and let the distance between both play a central role in our theorems. Thereby we extend the refined versions of the Bohr inequality for the class of the quasi-subordinations which contains both the classes of majorization and subordination as special cases. As a consequence, we prove Bohr type theorems for functions subordinate to convex or univalent functions.
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    Improved Bohr’s Inequality for Shifted Disks
    (01-03-2021)
    Evdoridis, Stavros
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    Rasila, Antti
    In this paper, we study the Bohr phenomenon for functions that are defined on a general simply connected domain of the complex plane. We improve known results of R. Fournier and St. Ruscheweyh for a class of analytic functions. Furthermore, we examine the case where a harmonic mapping is defined in a disk containing D and obtain a Bohr type inequality.
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    Integral transforms of functions with the derivative in a halfplane
    (01-01-1999) ;
    Rønning, F.
    Let A be the class of normalized analytic functions in the unit disk Δ and define the class Pβ = {f ∈ A| ∃α ∈ ℝ| Re{eia(f′(z) - β)} > 0, z ∈ Δ}. For a function f ∈ A the Alexander transform F0 is given by F0(z) - ∫10 f(tz)/tdt. Our main object is to establish a sharp relation between β and γ such that f ∈ Pβ implies that F0 is starlike of order γ, 0 ≤ γ ≤ 1/2. A corresponding result for the Libera transform F1(z) = 2 ∫10 f (tz)dt is also given.
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    The Quasiconvexity of Quasigeodesics in real normed vector spaces
    (10-03-2010)
    Huang, M.
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    Wang, X.
    In this paper, we assume that E is a real normed space of dimension at least two. The aim of this paper is to show that, for any convex domain in E, each quasigeodesic in this domain is quasiconvex in the norm metric. This result gives an affirmative answer to an open problem raised recently by Väisälä. © 2010 Birkhäuser / Springer Basel AG 2010.
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    Region of variability of univalent functions f(z) for which zf′(z) is spirallike
    (01-12-2008) ;
    Vasudevarao, A.
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    Yanagihara, H.
    For a complex number α with Re α > 0 let S α(λ) be the class of analytic functions f in the unit disk D with f(0) = 0 = f′(0) - 1, f″(0) = 2λe -1α cos α satisfying Reeiα (1 + f′(z)/zf″(z)) > 0 for z ∈ D. For z 0 ∈ D fixed, we determine the region of variability for log f′(z 0) when f ranges over the class Sα (λ). As a consequence, we obtain an estimate for a pre-Schwarzian norm for Sα(0). © 2008 University of Houston.
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    On Schwarz–Pick-Type Inequality and Lipschitz Continuity for Solutions to Nonhomogeneous Biharmonic Equations
    (01-06-2023)
    Li, Peijin
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    Li, Yaxiang
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    Luo, Qinghong
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    The purpose of this paper is to study the Schwarz–Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: Δ (Δ f) = g, where g : D¯ → C is a continuous function and D¯ denotes the closure of the unit disk D in the complex plane C. In fact, we establish the following properties for these solutions: First, we show that the solutions f do not always satisfy the Schwarz–Pick-type inequality 1-|z|21-|f(z)|2≤C,where C is a constant. Second, we establish a general Schwarz–Pick-type inequality of f under certain conditions. Third, we discuss the Lipschitz continuity of f, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric.
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    Schwarzian derivative and Landau's theorem for logharmonic mappings
    (01-08-2013)
    Mao, Zh
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    Wang, X.
    The main aim of this article is twofold. First, we introduce the Schwarzian derivative for logharmonic mappings F, and obtain several necessary and sufficient conditions for to be analytic. Second, we establish Schwarz' lemma for logharmonic mappings, through which two versions of the Landau's theorem for these functions are obtained. © 2013 Copyright Taylor and Francis Group, LLC.
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    The Bohr-Type Inequalities for Holomorphic Mappings with a Lacunary Series in Several Complex Variables
    (01-01-2023)
    Lin, Rouyuan
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    Liu, Mingsheng
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    In this paper, we mainly use the Fréchet derivative to extend the Bohr inequality with a lacunary series to the higher-dimensional space, namely, mappings from Un to U (resp. Un to Un). In addition, we discuss whether or not there is a constant term in f, and we obtain two redefined Bohr inequalities in Un. Finally, we redefine the Bohr inequality of the odd and even terms of the analytic function f so as to obtain conclusions for two different higher-dimensional alternating series.
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    Starlikeness and convexity of generalized bessel functions
    (13-08-2010)
    Baricz, à rpád
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    In this paper, we give sufficient conditions for the parameters of the normalized form of the generalized Bessel functions to be convex and starlike in the open unit disk. As an application of our main results, we solve a recent open problem concerning a subordination property of Bessel functions with different parameters. Moreover, we present a new inequality for the Euler gamma function, which we apply in order to have tight bounds for the generalized and normalized Bessel function of the first kind. © 2010 Taylor & Francis.