New posts in Integral Inequality

In which case these integrals are equal?

How prove this $\left(\int_{-\pi}^{\pi}f(x)\sin{x}dx\right)^2+\left(\int_{-\pi}^{\pi}f(x)\cos{x}dx\right)^2\le\frac{\pi}{2}\int_{-\pi}^{+\pi}f^2(x)dx$

Under what conditions do we have $\int_{0}^{\infty} |f(x)|^2 dx \leq C \int_{0}^{\infty} x^2 |f^{\prime}(x)|^2 dx$?

$\int_a^bf^2(x)\,dx\le \frac{2}{3}\int_a^bf(x)\,dx$ for a convex differentiable function

Proof of maximal principle on Laplace Equation involving Poisson integral formula

If $f^2(t) \le 1+2\int_0^tf(s)\mathrm{d}s$ prove $f(t)\le 1+t$

Does an integral inequality imply a pointwise inequality?

Integral inequality $\frac{1}{2M}\le\int_0^1xf(x)dx\le1-\frac{1}{2M}$

$f$ convex, $g$ concave and increasing, $\int_0^1 f = \int_0^1 g$, then $\int_0^1(f)^2 \geq \int_0^1(g)^2$

A tricky integral inequality