Problem 931
If \(\displaystyle f\) is bounded and concave down on \(\displaystyle [a,b]\), then \(\displaystyle (b-a){f(a)+f(b) \over 2} \ \leq\ \int_a^b f \ \leq\ (b-a)f\Big({a+b \over 2}\Big)\) Difficulty: 3. |
Problem 935
Let \(\displaystyle p,q>0\) and \(\displaystyle 1/p+1/q=1\). Show that for all \(\displaystyle x,y\geq 0\) \(\displaystyle xy\leq {x^p \over p} + {y^q \over q}.\) Difficulty: 3. |
Problem 936
Prove the following (a) If \(\displaystyle f,g:[a,b]\to\R\) are integrable, then \(\displaystyle \left(\int_a^b fg\right)^2 \le \left(\int_a^b f^2\right) \left(\int_a^b g^2\right) \) (Schwarz-inequality). (b) If \(\displaystyle f,g:[a,b]\to\R\) are integrable and \(\displaystyle p,q>0\) such that \(\displaystyle \frac1p+\frac1q=1\), then \(\displaystyle \int_a^b fg \le \left(\int_a^b |f|^p\right)^{1/p} \left(\int_a^b |g|^q\right)^{1/q} \) (Hölder-inequality). Difficulty: 3. |
Problem 932
Assume that \(\displaystyle f:[0,\infty)\to \R\) is strictly increasing continuous and \(\displaystyle f(0)=0\), \(\displaystyle \lim_{\infty}f=\infty\). Let \(\displaystyle g\) be the inverse function \(\displaystyle f\). Show that \(\displaystyle xy\leq \int_0^x f \ +\ \int_0^y g.\) Difficulty: 5. |
Problem 942
Prove that \(\displaystyle xy\leq (x+1)\log (x+1)-x+e^y-y-1\) holds for all pairs \(\displaystyle x,y\) of positive numbers. Difficulty: 5. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |