ABSTRACT
To obtain an upper bound, we can use Holder's inequality with p = q = 2 (see the table at the end of this section for an exact statement of the inequality). We have
206 IV Approximate Analytical Methods
[2) Cauchy-Schwartz-Bunyakowsky inequality (see Squire [19], page 21)
Equality occurs only when /(:z:) = kg(:z:), with k real. (3) Hardy's inequality (see Iyanaga and Kawada [9], page 1422)
(4) Modified Hardy's inequality (see Izumi and Izumi [8))
[5) Hardy-Littlewood supremum theorem (see Hardy, Littlewood, and Polya [7), page 298 ( #398))
46. Integral Inequalities 207
[6) Holder's inequality (see Squire [19), page 21)
11.• f(x)g(x) dx' $ (/.•1/(x)IP dx) 1
[7) Backward Holder's inequality (see Brown and Shepp [1))
s~p J [f(x -y)g(y)] dy $ (/1/(x)IP dx) 1/P (/lg(x)l' dx) 119 when f and g have compact support, p and q are positive, and 1/p+ 1/q = 1.
