Suppose that $f$ is a continuous function on $[0,1]$ and $$\int_0^x [f(t)]^2dt \le f(x) \quad \text{for all} \quad x \in[0,1].$$ Prove or disprove $$\min_{0\le x\le 1} f(x) \le 1.$$ In case the desired inequality does not hold, what is the best upper bound?
Thanks.
I assume that $f^2(t)$ means $\big(f(t)\big)^2$. I have a very weak bound $$\min_{x\in[0,1]}\,f(x)<2\sqrt{2}\,,$$ and do not know how to improve it. Maybe somebody can use my proof to get a better bound.
Suppose on the contrary that there exists a function $f:[0,1]\to\mathbb{R}$ satisfying $$\int_0^x\,\big(f(t)\big)^2\,\text{d}t\leq f(x)\text{ for all }x\in[0,1]\tag{*}$$ such that $$k:=\min_{x\in[0,1]}\,f(x)\geq 2\sqrt{2}\,.$$ From (*), we get that $$f(x)\geq \max\big\{k^2x,k\}\text{ for all }x\in[0,1]\,.$$ We use (*) once again and find that $$f(x)\geq \int_0^{\frac1k}\,k^2\,\text{d}x+\int_{\frac{1}{k}}^x\,(k^2t)^2\,\text{d}t=\frac{2k}{3}+\frac{k^4x^3}{3}$$ for $x\in\left[\dfrac1k,1\right]$.
Define polynomials $P_0$, $P_1$, $P_2$, $\ldots$ as follows: $$P_0(z):=1\,,\,\,P_1(z):=z\,,\,\,P_2(x):=\frac{2}{3}+\frac{z^3}{3}\,,$$ and $$P_r(z)=1+\int_{1}^z\,\big(P_{r-1}(\zeta)\big)^2\,\text{d}\zeta$$ for $r=3,4,5,\ldots$. It can be proven by induction that, for each $r=0,1,2,\ldots$, $$f(x)\geq k\,P_r(kx)$$ for all $x\in\left[\dfrac1k,1\right]$.
We can prove by induction that $0\leq P_r(z)\leq 1$ for every $z\in[0,1]$. That is, the constant term of $P_r$ is nonnegative for every $r=0,1,2,\ldots$. It is obvious that the coefficients of higher-order terms in $P_r$ are nonnegative. Furthermore, the degree of $P_r$ is $2^r-1$, and the coefficient $\lambda_r$ of the $(2^r-1)$-st degree term of $P_r$ is given by the recurrence relation $$\lambda_r=\frac{1}{2^r-1}\lambda_{r-1}^2\,.$$ In other words, $$\lambda_r=\frac{1}{\prod\limits_{j=1}^r\,(2^j-1)^{2^{r-j}}}\geq \frac{1}{\prod\limits_{j=2}^r\,2^{j\cdot2^{r-j}}}=\frac{1}{2^{3\cdot 2^{r-1}-r-2}}\,.$$ That is, $$f(1)\geq k\,P_r(k)\geq \frac{k^{2^r}}{2^{3\cdot2^{r-1}-r-2}}$$ for every $r=0,1,2,\ldots$. However, as $k\geq 2\sqrt{2}$, $k^{2^r}$ grows faster than $2^{3\cdot2^{r-1}-r-2}$, namely, $$\lim_{r\to\infty}\,\frac{k^{2^r}}{2^{3\cdot2^{r-1}-r-2}}=\infty\,.$$ This yields a contradiction.
Remark. I believe that the polynomials $P_r$ converge pointwise, as $r\to\infty$, to $P$ on $(-2,+2)$, where $$P(z):=\frac{1}{2-z}\text{ for }z\in(-2,+2)\,.$$ I conjecture also that, for $z\geq 2$, $\lim\limits_{r\to\infty}\,P_r(z)=\infty$. If this is true, then it follows immediately that $\min\limits_{x\in[0,1]}\,f(x)<2$.
Counterexample. Interestingly, let $k\in(1,2)$ and define $$f(x):=\max\left\{k,\frac{k}{2-kx}\right\}\text{ for all }x\in[0,1]\,.$$ Then, we see that $\min\limits_{x\in[0,1]}\,f(x)=k$. Furthermore, $$f(x)=k\geq k^2x=\int_0^x\,k^2\,\text{d}t=\int_0^x\,\big(f(t)\big)^2\,\text{d}t$$ for $x\in\left[0,\dfrac1k\right]$. For $x\in\left[\dfrac1k,1\right]$, we have $$f(x)=\frac{k}{2-kx}=k+\int_{\frac1k}^x\,\left(\frac{k}{2-kt}\right)^2\,\text{d}t=\int_0^x\,\big(f(t)\big)^2\,\text{d}t\,.$$ This shows that, if $\min\limits_{x\in[0,1]}\,f(x)<c$ for every such function $f$, then $c\geq 2$.
We can show that $$ \min_{x\in[0,1]}\,f(x) < 2 \, , $$ and that is the best possible result, as Batominovski demonstrated with his counter-example.
Assume that $f(x) \ge 2$ in the interval. For $0 < x \le 1$ we define $$ g(x) = \frac{1}{\int_0^x f^2(t) \, dt} \, . $$ Then $$ g'(x) = -\frac{f^2(x)}{\left( \int_0^x f^2(t) \, dt\right)^2} \le -1 $$ and therefore $$ g(x) \ge g(1) + (1 - x) > 1 - x \, . $$ This implies $$ 4 x \le \int_0^x f^2(t) \, dt = \frac{1}{g(x)} < \frac{1}{1-x} $$ and setting $x = \frac 12$ gives a contradiction.
