In this paper, we obtain an extension of multivariable integral inequality of Hilbert-Pachpatte type. By specializing the upper estimate functions in the hypothesis and the parameters , we obtain many special cases. Hilbert's double series theorem [3, p. 226] was proved first by Hilbert in his lectures on integral equations. The determination of the constant, the integral analogue, the extension, other proofs of the whole or of parts of the theorems and generalizations in different directions have been given by several authors (cf. [3, Chap. 9]). Specifically, in [10] – [14] the author has established some new inequalities similar to Hilbert's double-series inequality and its integral analogue which we believe will serve as a model for further investigation. Recently, G. [2] established a new class of related integral inequalities from which the results of Pachpatte [12] – [14] are obtained by specializing the parameters and the functions Φ i. A representative sample is the following. for the careful reading of the manuscript which led to a considerable improvement in the presentation of this paper.
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