## 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

### Keywords:

inequalities; trigonometric functions; convex functions