[PDF] Top 20 Monotonicity of a mean related to polygamma functions with an application

... f is called logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and its logarithm ln f satisfies (–) n (ln f (x)) (n) ≥  for all n ∈ N on I (see []). For convenience, we ... See full document

... generalized functions, and sug- gested that many properties of classical gamma, digamma and polygamma functions have a counterpart in this more general ...classical functions to the k-gamma, ... See full document

... 5. Alzer, H., Wells, J.: Inequalities for the polygamma functions. SIAM J. Math. Anal. 29(6), 1459–1466 (1998) 6. Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. ... See full document

... 13. Batir, N: Sharp bounds for the psi function and harmonic numbers. Math. Inequal. Appl. 14(4), 917-925 (2011) 14. Guo, B-N, Qi, F: Sharp inequalities for the psi function and harmonic numbers. Analysis 34(2), 201-208 ... See full document

... the polygamma functions. The gamma, digamma and polygamma functions play an im- portant role in the theory of special functions, and have many applications in other many branches, such ... See full document

... the polygamma functions is proved to be completely ...the monotonicity and convexity of a function originated from establishing the best upper and lower bounds in Kershaw’s inequality is ... See full document

... and polygamma functions, and other analytic ...completely monotonicity re- sults related to Ψ s,t (x) and Gautschi-Kershaw inequalities (1) and (2) are ... See full document

... for x ∈ I and k ∈ N , where f (0) (x) means f (x) and N stands for the set of all positive integers. See [14, Chapter XIII], [31, Chapter 1], or [33, Chapter IV]. The class of completely monotonic functions may be ... See full document

... n – 1, n + 1) for various c has been determined in [12, Theorem 1]. We also note that the logarithmically complete monotonicity of ratios of two gamma functions is closely related to the divided ... See full document

... p, q ∈ R , a > , b >  } and strictly increasing with respect to its parameters p, q ∈ R for fixed a, b >  with a = b. Many bivariate means are particular cases of the Stolarksy mean, and many remarkable ... See full document

... used monotonicity rule in elementary calculus is that f is increasing (decreasing) on [a, b] if f : [a, b] → R is continuous on [a, b] and has a positive (negative) derivative on (a, b), and it can be proved ... See full document

... 1.. The proof is complete. By Lemma 1 and Lemma 2, the following corollary is obvious.. The proof is complete.. The proof is complete. Stolarsky, Generalizations of the logarithmic mean,[r] ... See full document

... Recently, Sarikaya et al. [10] introduced a new analogue of the Gamma func- tion which they called the conformable Gamma function. Motivated by their results, we have established some inequalities involving the ... See full document

... numerous functions, which are defined in terms of gamma, polygamma, and other special functions, are (logarithmically) completely monotonic and used this fact to derive many interesting new ... See full document

... called polygamma functions. These functions play an important role in various branches of mathematics as well as in physics and engi- ...these functions, please refer to [], ... See full document

... Yang, Z-H: Very accurate approximations for the elliptic integrals of the second kind in terms of Stolarksy means.. Yang, Z-H, Chu, Y-M, Zhang, W: Accurate approximation for the complete[r] ... See full document

... Gini mean is also called two-parameter arithmetic mean, F1, 0; r, s; a, b is just the two-parameter mean or extended mean or Stolarsky mean is also called two-parameter logarithmic ... See full document

... gamma functions, please refer to the papers [11, 12, 14, 15, 21, 24, 25, 26, 29, 30, 32, 33, 35, 36, 37, 40, 41, 45, 46], the expository and survey articles [27, 28, 38, 39] and closely related references ... See full document

... we mean that the constant √ e/π in () cannot be replaced by a number which is greater than √ e/π and the constant / in () cannot be replaced by a number which is less than ... See full document

... Cooke’s proof is rather complicated as it depends on some delicate estimates involving the Lommel functions and several properties of Bessl functions In [3], Makai proved this result for[r] ... See full document