A Unitary Treatment of Certain Inequalities Involving Means

The aim of this paper is to state and prove certain inequalities that involve means (e.g., the arithmetic, geometric, logarithmic means) using a particular result. First of all we recall useful properties of a real-valued convex function that will be used in the proof of our inequalities. Further, we present three inequalities, the first involving the logarithmic mean, the second involving the classical arithmetical and geometrical means and in the last we introduce a new mean. Finally, we give alternate proofs to the Schweitzer’s inequality and Khanin’s inequality.

Maximal summability operators on the dyadic hardy spaces

It is proved that the maximal operators of subsequences of N?rlund logarithmic means and Ces?ro means with varying parameters of Walsh-Fourier series is bounded from the dyadic Hardy spaces Hp to Lp. This implies an almost everywhere convergence for the subsequences of the summability means.

SOME PROPERTIES OF BLASCHKE TYPE PRODUCTS FOR THE HALF-PLANE

In this paper we obtain balance formulas for the logarithmic means of Blaschke type functions and investigate their boundary values.

Some New Inequalities Involving κ-Fractional Integral for Certain Classes of Functions and Their Applications

In this article, we present several new inequalities involving the κ-fractional integral for the integrable function ℱ which satisfies one of the following conditions: aℱq is preinvex for some q>1; bℱ′ is bounded; cℱ′ is a Lipschitz function. As applications, we establish new inequalities for the weighted arithmetic and generalized logarithmic means.

On the almost sure convergence of randomly indexed maximum of random variables

We prove an almost sure random version of a maximum limit theorem, using logarithmic means for \(\max_{1\leq i\leq N_n} X_i\), where \(\{X_n, n \geq 1\}\) is a sequence of identically distributed random variables and \(\{N_n, n \geq 1\}\) is a sequence of positive integer random variables independent of \(\{X_n, n \geq 1\}\). Furthermore, we consider the almost sure random version of a limit theorem for \(k\)th order statistics.

ON INTEGRAL LOGARITHMIC MEANS OF BLASCHKE PRODUCTS FOR A HALF-PLANE

Using the Fourier transforms method for meromorphic functions we characterize the behavior of the integral logarithmic mean of arbitrary order of Blaschke products for the half-plane.