Zmorovič’s and Related Inequalities

Abstract

In [1] V. A. Zmorovič has proved the following theorem: Theorem 1. If the function f: [a-h a+h] → R is twice continuously-differentiable, then with

$$\int\limits_{a - h}^{a + h} {{{(f''(x))}^2}dx \geqslant \frac{3}{{2{h^3}}}{{[f(a + h) - 2f(a) + f(a - h)]}^2},}$$

(1.1)

equality it and only if f is given by

$$f(x) = \left\{ {\begin{array}{*{20}{c}}{{C_1}\{ {{(h - a + x)}^3} + 6{h^2}(a - x)\} + {C_2}x + {C_3} (x \in [a - h,a]),} \\{{C_1}{{(h + a - x)}^3} + {C_2}x + {C_3} (x \in [a,a + h]),}\end{array}} \right.$$

where C ₁, C ₂, C ₃ are arbitrary real constants.