Sentence examples for hardy function from inspiring English sources

For the Hardy function per H n ( x, α ), Wen and Wang in [2] (see Corollary 1 in [2]) obtained the following result: Let x, y ∈ ( 0, ∞ ) n, α ∈ n.

Obviously, the Hardy function per H n ( x, α ) is a symmetric function.

In this paper, we establish the following Minkowski-type inequality (5) involving Hardy function.

For the Hardy function, we have the following well-known Hardy inequality (see[3, 7]): Let α, β ∈ n.

Due to the facts that the symmetric polynomial and certain symmetric functions can be expressed by the Hardy function (see [6] and Remark 1), and that the interpolating quasi-polynomial can be expressed by the generalized Vandermonde determinant (see [4, 5]), the Hardy function and the generalized Vandermonde determinant are of great significance in mathematics.

The Hardy matrix H n ( x, α ), the Hardy function per H n ( x, α ) and the generalized Vandermonde determinant det H n ( x, α ) are defined in this paper.

An important problem in analytic number theory is to gain an understanding of the moments of the Hardy -function function and the moments of its derivative which are defined by (1.5).

It is known that the behavior of on the critical line is reflected by the Hardy -function as a function of a real variable, defined by (1.4).

(4.8) By the definition of the Hardy-Littlewood function, we obviously deduce limsup_{krightarrowinfty} M^{k}f(x le|f |_{infty}.

This paper is mainly devoted to the study of the Hardy-Littlewood maximal function on noncommutative Lorentz spaces and to obtaining ( p, q ) - ( p, q ) -type inequality for the Hardy-Littlewood maximal function on noncommutative Lorentz spaces.

Finally, we investigate the properties of the iterated Hardy-Littlewood maximal function.