Does $2\int_{0}^{y^{2}}c(t)\,dt<3\frac{1}{y}\int_{0}^{y^{2}}\sqrt{t}c(t)\,dt$ always hold for positive and increasing $c(t)$? I come across one problem. Does $2\int_{0}^{y^{2}}c(t)\,dt&lt;3\frac{1}{y}\int_{0}^{y^{2}}\sqrt{t}c(t)\,dt$ always hold ? The $y$ is any positive number. The domain of $c(t)$ is $[0,\infty)$ and it is smooth, strictly increasing the positive. I think it is not true. But I cannot find out one counter-example of $c(t)$. Can you help ? Thank you.
The inequality is equivalent to 

$$  2y \int_0^{y^2} c(t) dt \le 3 \int_0^{y^2}\sqrt{t}  c(t) dt.$$ 

Let's prove this by differentiation. 

$$ \frac{d}{dy} \left (  2y \int_0^{y^2} c(t) dt\right)  = 2 \int_0^{y^2} c(t) dt + 4y^2 c(y^2) &lt; 6 y^2 c(y^2),$$ 

because $c$ is strictly increasing. Also, 

$$ \frac{d}{dy}\left (  3\int_0^{y^2} \sqrt{t} c(t) dt\right) = 6 y^2 c(y^2),$$

and result follows, because both are equal to zero at $y=0$.
If $c(t)=1$ we have

$2\int_{0}^{y^{2}}c(t)\,dt=3\frac{1}{y}\int_{0}^{y^{2}}\sqrt{t}c(t)\,dt$

Edit: this is no counterexample since $c$ shoul be increasing

FRED