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The sharpness of some recent integral inequalities is discussed and the corresponding extremal functions are pointed out. It is also proved that the cases of equality can equivalently be obtained by solving an integral equation of Volterra type with discontinous kernel χA (t) . This integral equation of independent interest is solved for every measurable set A on (a, b) ,−∞ < a < b < ∞ . Mathematics subject classification (2000): 26D15, 27D07.

On sharpness of some integral inequalities and an integral equation of Volterra type