On some inequalities for the gamma and psi functions

We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and super-additive functions which are related to and .

     

An extension of the incomplete beta function for negative integers

The incomplete beta function Bx(a,b) is defined for a,b>0 and 0<x<1. Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function Bx(a,b) for negative integer values of a and b. Further we prove that the function exists for m,n=0,1,2,… and all a and b.     

指数凸函数的积分不等式及其在Gamma函数中的应用

仿对数凸函数的概念,给出指数凸函数的定义,并证明有关指数凸函数的几个积分不等式,作为应用,得到一个新的Kershaw型双向不等式.     

Nikolskii-type inequalities for shift invariant function spaces

Let be a vectorspace of complex-valued functions defined on of dimension over . We say that is shift invariant (on ) if implies that for every , where on . In this note we prove the following.

Theorem. Let be a shift invariant vectorspace of complex-valued functions defined on of dimension over . Let . Then

for every and

     

Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [a,b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.     

Inequalities for the double gamma function

We establish various new upper and lower bounds in terms of the classical gamma and digamma functions for the double gamma function (or Barnes G-function).     

Some inequalities for the gamma function

Several inequalities involving gamma function are obtained. They are established using elementary properties of logarithmically convex functions.     

Inequalities for the gamma function

We prove the following two theorems:

(i) Let be the th power mean of and . The inequality

holds for all if and only if , where denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997).

(ii) The inequalities

are valid for all if and only if and , while holds for all if and only if and . These bounds for improve those given by G. D. Anderson an S.-L. Qiu (1997).

     

New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

On some properties of the Gamma function

Anderson and Qiu (1997) conjectured that the function is concave for 1$">. In this paper we prove this conjecture. We also study the monotonicity of some functions connected with the psi-function and derive inequalities for and .

     

Monotonicity properties and functional inequalities for the Volterra and incomplete Volterra functions

In this paper we prove some monotonicity, log–convexity and log–concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some functional inequalities (like Turán type inequalities) as well as we determined sharp upper and lower bounds for the normalized incomplete Volterra functions in terms of weighted power means.     

On some properties of the gamma function

In this paper we prove a complete monotonicity theorem and establish some upper and lower bounds for the gamma function in terms of digamma and polygamma functions.     

On some pointwise inequalities

In a recent paper (Studia Math. 138 (2000) 285-291) we proved pointwise estimates relating some classical maximal and singular integral operators. Here we show that, in a sense, there are more flexible inequalities which not only imply the previously known results but also give something new. In particular, they hold for the multilinear Calderón-Zygmund operators. This result gives a new approach to a recent work by Grafakos and Torres, and unifies some classical inequalities by Cotlar and Coifman and Fefferman.     

A characterization of some weighted inequalities for the vector-valued weighted maximal function

We give in this paper a necessary and sufficient condition of weighted weak and strong type norm inequalities for the vector-valued weighted maximal function.     

A monotonicity property of the composition of regularized and inverted-regularized gamma functions with applications

The gamma function and its various modifications such as the (upper) incomplete, regularized and inverted-regularized incomplete gamma functions are of importance in both theory and applications. In this note we observe an ‘if and only if ’ relationship between a certain axiom of insurance risk management and a monotonicity property of the composition of regularized and inverted-regularized incomplete gamma functions, assuming that the risks follow gamma distributions. We derive the monotonicity property by utilizing the above noted relationship and a probabilistic technique. The aforementioned insurance axiom, called consistent no-undercut, is explained in detail and related to several techniques of analysis.     

A monotoneity property of the gamma function

In this paper we obtain a monotoneity property for the gamma function that yields sharp asymptotic estimates for as tends to , thus proving a conjecture about .