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In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor F distribution.

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In this paper we present some Turán type inequalities for the probability density function (pdf) of the non-central chi-squared distribution, non-central chi distribution and Student distribution, respectively. Moreover, we improve a result of Laforgia and Natalini concerning a Turán type inequality for the modified Bessel functions of the second kind.

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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 min j 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 , Á. , Turán type inequalities for regular Coulomb wave functions , J. Math. Anal. Appl., 430 ( 1 ) ( 2015 ), 166 – 180 . 10.1016/j.jmaa.2015.04.082 [10] Baricz , Á. , Çálar , M. , Deniz , E. and Toklu , E. , Radii of starlikeness

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