×

Two inequalities. (English) Zbl 0841.26009

The author proves that \[ \sin^{- 1} x\leq \pi x/(2+ (1- x^2)^{1/2}),\qquad 0\leq x\leq 1, \] and \({\pi\over 4} \int^1_{- 1} f(x+ vt)\cos (\pi t/2) dt\leq (f(x+ v)+ f(x- v))/2\), \(x,v\in \mathbb{R}\), where \(f> 0\) and \(\log f\) is convex on \(\mathbb{R}\).

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26A51 Convexity of real functions in one variable, generalizations
26D15 Inequalities for sums, series and integrals
PDF BibTeX XML Cite