Minkowski Inequality

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Minkowski Inequality


an inequality of the form

where ak and ak (k = 1,2, . . . , n) are non-negative numbers and r > 1. The Minkowski inequality has analogs for infinite series and integrals. The inequality was established by H. Minkowski in 1896 and expresses the fact that in n-dimensional space, where the distance between the points x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn) is given by

the sum of the lengths of two sides of a triangle is greater than the length of the third side.

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The classical Holder's and Minkowski's inequalities are usually defined as follows.
In the present work, our objective is to provide some generalized Holder's and Minkowski's inequalities for Jackson's q-integral.
By using the mean value theorem and using Holder's and Nirenberg's, Minkowski's inequalities for integrals, Fourier multiplier theorems for operator-valued functions in [X.sub.p] spaces and Young's inequality, we obtain