Can this inequality hold?

Given $$\Omega$$ a compact subset of $$\mathbb{R}^n$$ and $$f\in H^1(\Omega,\mathbb{C})$$ with zero average, I wonder if there is an inequality of the form $$\int_\Omega \phi(|f|^2)\varphi(|f|^2)\ dx \leq C\left( \int_\Omega \phi(|f|^2)\ dx \right)^{1/a} \left(\int_\Omega \varphi(|f|^2)\ dx \right)^{1/a}$$ where $$a>1$$, $$C>0$$ (might depend on $$\Omega$$) and $$\varphi,\phi$$ are real-valued and convex functions. Any idea is welcome, thanks for advance.

I checked this for the simple case of  $$\varphi,\phi$$ being constants and it works (it is enough to take $$C\geq 1$$).

Indeed, suppose that $$\phi(u)=\varphi(u)=u$$ for all $$u$$. Let $$f=tg$$, where $$t\in(0,\infty)$$ and $$g$$ is a function in $$H^1(\Omega,\mathbb{C})$$ with zero average and such that $$\int_\Omega |g|^4\,dx>0$$. Then the left-hand side of your inequality is $$t^4\int_\Omega |g|^4\,dx$$ and the right-hand side is $$Ct^{4/a}(\int_\Omega |g|^2\,dx)^{2/a}$$, which is less than the left-hand side if (i) $$a>1$$ and $$t$$ is large enough or (ii) $$0 and $$t$$ is small enough.
Finally, this ineqiality cannot hold in general even for $$a=1$$. E.g., suppose that $$\Omega=[-1,1]^n$$, $$\phi(u)=\varphi(u)=u$$ for all $$u$$ (as before), and $$f(x)=x_1^{2m+1}$$ for natural $$m$$ and $$x=(x_1,\dots,x_n)\in\Omega$$. Then the left-hand side of your inequality is $$2^n/(8m+5)$$, and the right-hand side is $$C\,2^{2n}/(4m+3)^2$$, which is less than the left-hand side if $$m$$ is large enough.
So, your inequality fails to hold in general for any given real $$a>0$$.
• But althoug $t$ is large and K is small, the inequality holds for some small enough $a$, am I wrong? I mean, $$t^{4-4/a}\leq CK^{2/a-1}$$ holds true for some small $a$ and large enough C, – R. N. Marley Sep 17 '19 at 22:18
• When I say small $a$, I mean $a$ close to $1$ in the previous comment. – R. N. Marley Sep 17 '19 at 22:26
• @R.N.Marley : In your question, $a>1$. However, as now shown, your inequality cannot hold in general for any given real $a>0$. – Iosif Pinelis Sep 18 '19 at 2:51