[PDF] Top 20 The monotonicity and convexity of a function involving psi function with applications

... 22. Batir, N: Sharp bounds for the psi function and harmonic numbers. Math. Inequal. Appl. 14, 917-925 (2011) 23. Batir, N: Some new inequalities for gamma and polygamma functions. J. Inequal. Pure Appl. ... See full document

... factorial function (a, n) ≡ a(a + ) · · · (a + n – ), n = , , ...the function F(a, b; c; x) has many important applications in geometric function theory, theory of mean values, and several ... See full document

... functions, psi function, polygamma function, divided difference, monotonicity, logarithmic convexity, completely mono- tonic function, logarithmically completely monotonic ... See full document

... a monotonicity property for the function involving the strongly convex function, prove the classical majorization theorem for majorized n-tuples by using strongly convex functions, give some ... See full document

... There is a rich literature on refinements, extensions, and generalizations of the inequalities in (4), for examples, [2, 8, 9, 13, 19] and references therein. Note that the inequalities in (4) are direct consequences of ... See full document

... gamma function Γ(x) is one of the most important special functions and has much extensive applications in many branches, for example, statistics, physics, engineering, and other mathematical ... See full document

... A function involving the divided differences of the psi function and the polygamma functions is proved to be completely ...the monotonicity and convexity of a function ... See full document

... monotonic function of second order in (0, ...convex function in [0, ∞), then the function [f(x)] f(ax) a for a ≥ 1 (or 0 < a ≤ 1 respectively) is decreasing (or increasing respectively) in [0, ... See full document

... properties involving the gamma, digamma, and polygamma functions can be found in some books ...as monotonicity, convexity, and complete monotonicity yield numerous inequalities related to ... See full document

... functions involving gamma and q-gamma functions, Nonlinear ...Qi, Monotonicity and logarithmic convexity for a class of elementary functions ... See full document

... J. Chen, A complete monotonicity property of the gamma function, J. Guo, Complete monotonicities of functions involving the gamma and digamma. functions, RGMIA Res. Chen, Some completely[r] ... See full document

... the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric ...As applications, we find several complete monotonicity properties for the functions ... See full document

... gamma function and psi function ψ have been the subject of intensive ...and monotonicity proper- ties for these functions can be found in the literature ... See full document

... [30] F. Qi and B.-N. Guo, An inequality between ratio of the extended logarithmic means and ratio of the exponential means, Taiwanese J. Math. 7 (2003), no. 2, 229–237. RGMIA Res. Rep. Coll. 4 (2001), no. 1, Art. 8, ... See full document

... gamma, psi and polygamma functions, and other analytic techniques, this note provides an alternative proof of a monotonicity and convexity property by ... See full document

... 6. Smith, WD: The gamma function revisited. http://schule.bayernport.com/gamma/gamma05.pdf (2006) 7. Nemes, G: New asymptotic expansion for the gamma function. Arch. Math. (Basel) 95, 161-169 (2010) 8. ... See full document

... functions involving R(x) = ψ (x + 1/2) – ln x with a parameter are completely monotonic on (0, ∞), find three new sequences which are fast convergence toward the Euler-Mascheroni constant, and give a positive ... See full document

... and psi functions have many applications in the areas of mathematics, physics, and engineering ...and psi functions have attracted the interest of many ...the psi function ψ (x) can be ... See full document

... complete monotonicity of a class of functions involving the Euler’s gamma function are proved, a class of the first Kershaw type double inequalities are established, and the first Kershaw’s double ... See full document

... Schur convexity was introduced by Schur in 1923 6, and it has many important applications in analytic inequalities 7–12, linear regression 13, graphs and matrices 14, combinatorial optimization 15, ... See full document