Probability theoretic generalizations of Hardy’s and Copson’s inequality

Generalizations of Hölder’s Inequality

Some generalized Hölder’s inequalities for positive as well as negative exponents are obtained. 2000 Mathematics Subject Classification. 26D07, 26D15.

Generalizations of Popoviciu’s inequality

We establish a general criterion for inequalities of the kind convex combination of f (x1) , f (x2) , ..., f (xn) and f (some weighted mean of x1, x2, ..., xn) ≥ convex combination of f (some other weighted means of x1, x2, ..., xn) , where f is a convex function on an interval I ⊆ R containing the reals x1, x2, ..., xn, to hold. Here, the left hand side contains only one weighted mean, while t...

The Generalizations of Hilbert's Inequality

Two Generalizations of Landau’s Inequality

Let π(x) be the number of prime numbers not exceeding x . In the introduction we present some inequalities related to this function; these have been presented in several articles about the Number Theory. In the present paper we obtain two inequalities which generalize Landau’s Inequality, π(2x) 2π(x) for any integer x 2 . Also, we obtain the inequality 2[xπ(x) + yπ(y)] > (x + y)π(x + y) , for a...

Some refinements and generalizations of Carleman's inequality

The constant is the best possible. There is a vast literature which deals with alternative proofs, various generalizations and extensions, and numerous variants and applications in analysis of inequality (1.1); see [1, 2, 3, 5, 7, 9, 8, 10, 13, 14, 15, 16, 17, 18, 19] and the references given therein. According to Hardy (see [6, Theorem 349]), Carleman’s inequality was generalized as follows. I...