In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams’ results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels sati...

In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams’ results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels sati...

In this paper, we present weighted integral inequalities of Hermite-Hadamard type for differentiable preinvex and prequasiinvex functions. Our results, on the one hand, give a weighted generalization of recent results for preinvex functions and, on the other hand, extend several results connected with the Hermite-Hadamard type integral inequalities. Applications of the obtained results are prov...

In this paper we consider two extremal problems for algebraic polynomials in L 2 metrics. (1) Let Pn be the class of all algebraic polynomials P(x) = akxk of degree at most nand IJPllda= (fIR IP(x)1 2 da(x))1/2, where da(x) is a nonnegative measure on lit We determine the best constant in the inequality lakl:::; Cn,k(da)IJPl!da-, for k = O,l, ... ,n, when P E Pn and such that = 0, k = 1, ... ,m...

Integral means inequalities are obtained for the fractional derivatives of order p + λ (0 5 p 5 n; 0 5 λ < 1) of functions belonging to certain general subclasses of analytic functions. Relevant connections with various known integral means inequalities are also pointed out.

Nonlinear integral inequalities of Gronwall-Bihari type for piecewise continuous functions are solved. Inequalities for functions with delay as well as functions without delays are considered. Some of the obtained results are applied in the deriving of estimates for the solutions of impulsive integral, impulsive integro-differential and impulsive differential-difference equations.

Using the integral expression of Mathieu's series and some integral and analytic inequalities involving periodic functions and the generating function of Bernoulli numbers, we present several new inequalities and estimates for Mathieu's series and generalize Math-ieu's series. Two open problems are proposed.

The unification and extension of differential equations, difference equations, q-difference equations, and so on to the encompassing theory of dynamic equations on time scales was initiated by Hilger 1 in his Ph.D. thesis in 1988. During the last few years, some integral inequalities on time scales related to certain inequalities arising in the theory of dynamic equations had been established b...