Landau-Kolmogorov inequality

Let $f$ be a $C^{2}$ function on the real line. Assume $f$ is bounded with bounded second derivative. Let

Prove that

Let $f(x) \ (-\infty <x<+\infty )$ be a twice differentiable function, and

Prove the inequality:

Let $f(x) \in C^{(2)}[0 ; 1]$, $f(0) = f(1) = 0$, and suppose there exists a number $M$ such that for all $x \in (0 ; 1)$ the inequality $\left|f^{\prime \prime }(x)\right| \leqslant M$ holds. Prove that

Let the function $f(x)$ be twice differentiable on $\mathbb {R}$, and let $M_{k} = \sup_{x \in \mathbb {R}}\left|f^{(k)}(x)\right| < +\infty \quad (k = 0,1,2)$. Prove that $M_{1}^{2} \leqslant 2 M_{0} M_{2}$.

Let $f \in C^{2}((0,1))$ and $f(x)=o(1)$, $f^{\prime \prime }(x)=O\left(x^{-2}\right)$ as $x \rightarrow +0$. Prove that $f^{\prime }(x)=o\left(x^{-1}\right)$ as $x \rightarrow +0$.

Let $\Delta$ be an interval, $f \in C^{2}(\Delta )$, $M_{k}=\sup_{\Delta }\left|f^{(k)}\right|, k=0,1,2$.

Let $f \in C^{2}([0,+\infty ))$ and $\int_{0}^{+\infty }\left|f(x)\right| d x<+\infty$. Prove the inequalities:

Suppose $a \in \mathbb {R}^{1}$, $f$ is a twice-differentiable real function on $(a, \infty )$, and $M_{0}, M_{1}, M_{2}$ are the least upper bounds of $\left|f(x)\right|,\left|f^{\prime }(x)\right|,\left|f^{\prime \prime }(x)\right|$, respectively, on $(a, \infty )$. Prove that

Hint: If $h > 0$, Taylor's theorem shows that

for some $\xi \in (x, x+2 h)$. Hence

To show that $M_{1}^{2} = 4 M_{0} M_{2}$ can actually happen, take $a = -1$, define

and show that $M_{0} = 1, M_{1} = 4, M_{2} = 4$. Does $M_{1}^{2} \leq 4 M_{0} M_{2}$ hold for vector-valued functions too?

Suppose $f$ is twice-differentiable on $(0, \infty ), f^{\prime \prime }$ is bounded on $(0, \infty )$, and $f(x) \rightarrow 0$ as $x \rightarrow \infty$. Prove that $f^{\prime }(x) \rightarrow 0$ as $x \rightarrow \infty$.

Hint: Let $a \rightarrow \infty$ in Exercise 15.