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Mathematical Functions with Formulas , Graphs and Mathematical Tables, Dover Publications, New York, 1965 . [2] B aricz , Á. , Turán type inequalities for generalized

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Baricz, Á. , Turán type inequalities for generalized complete elliptic integrals, Mathematische Zeithschrift , 256 (2007), no. 4, 895–911. MR 2008f :26016 Baricz Turán

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Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min [5] to include L p spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.

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Abstract  

Turán’s book [2], in Section 19.4, refers to the following result of Gábor Halász. Let a 0, a 1, ..., a n−1 be complex numbers such that the roots α 1, ⋯, α n of the polynomial x n + a n−1 x n−1 + ⋯ + a 1 x + a 0 satisfy minj Re α j ≧ 0 and let function Y(t) be a solution of the linear differential equation Y (n) + a n−1 Y (n−1) + ⋯ + a 1 Y′ + a 0 Y = 0. Then

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$|Y(0)| \leqq cn^5 \int_0^1 {|Y(t)|dt} .$$ \end{document}
((1)) In particular, (1) holds for polynomials of degree at most n − 1 and functions of the form
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Y(t) = \sum\limits_{j = 1}^n {b_j e^{\alpha _j t} }$$ \end{document}
where b 1,..., b n are arbitrary complex numbers and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\min }\limits_j$$ \end{document}
Re α j ≧ 0. In this paper we improve the exponent 5 on the right-hand side to the best possible value (which is 2) and prove an analogous inequality where the integration domain is symmetric to the origin.

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Baricz, A. , Turán type inequalities for some probability density functions, Stud. Sci. Math. Hung. , 47 (2) (2010), 175–189. Baricz A

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] Baricz , Á. , Geometric properties of generalized Bessel functions , Publ. Math. De brecen , 73 ( 2008 ), 155 – 178 . [9] Baricz , Á. , Turán type inequalities for regular Coulomb wave functions , J. Math. Anal. Appl., 430 ( 1 ) ( 2015

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