Abstract
In this paper, Jensen’s inequality and Fubini’s Theorem are extended for the function of several variables via diamond integrals of time scale calculus. These extensions are used to generalize Hardy-type inequalities with general kernels via diamond integrals for the function of several variables. Some Hardy Hilbert and Polya Knop type inequalities are also discussed as special cases. Classical and new inequalities are deduced from the main results using special kernels and particular time scales.
Highlights
Most of the inequalities relative to the integration and differentiation have their analog for sums and differences
In 1988, Stephan Hilger introduced the theory of time scales that provides a platform to deal with discrete and continuous cases together [1,2]
Jensen’s inequality for the diamond integral via the function of several variables is being proved
Summary
Most of the inequalities relative to the integration and differentiation have their analog for sums and differences. H. Hardy has introduced the following integral inequality ([3], Theorem 327): Rr. If p > 1, g(r ) ≥ 0 and G (r ) = 0 g(s)ds, :. In [4], Hardy proved the following generalization of inequality (1). In [19], Nosheen et al studied Hardy-type inequalities for functions of several variables with general kernels using the delta-integral. Anastassiou generalized Hardy-type inequalities with general kernel, using Diamond-α integral, are given as: Consider a continuous weight function w : [ a, b]T1 → R+ , and the function: v(s) :=. Hardy-type inequalities via diamond integrals using a multivariable convex function with general kernels are proved. Multiplicative, reflexive and monotonicity properties of -integral, see [22]
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