91Ó°ÊÓ

Let \(f\) be defined for all real \(x\), and suppose that $$ |f(x)-f(y)| \leq(x-y)^{2} $$ for all real \(x\) and \(y\). Prove that \(f\) is constant.

Suppose \(f\) and \(g\) are complex differentiable functions on \((0,1), f(x) \rightarrow 0, g(x) \rightarrow 0\), \(f^{\prime}(x) \rightarrow A, g^{\prime}(x) \rightarrow B\) as \(x \rightarrow 0\), where \(A\) and \(B\) are complex numbers, \(B \neq 0 .\) Prove that $$ \lim _{x \rightarrow 0} \frac{f(x)}{g(x)}=\frac{A}{B} $$ Compare with Example 5.18. Hint: $$ \frac{f(x)}{g(x)}=\left\\{\frac{f(x)}{x}-A\right\\} \cdot \frac{x}{g(x)}+A \cdot \frac{x}{g(x)} $$ Apply Theorem \(5.13\) to the real and imaginary parts of \(f(x) / x\) and \(g(x) / x\).

Suppose \(f\) is defined in a neighborhood of \(x\), and suppose \(f^{\prime \prime}(x)\) exists. Show that $$ \lim _{x \rightarrow 0} \frac{f(x+h)+f(x-h)-2 f(x)}{h^{2}}=f^{\prime \prime}(x) $$ Show by an example that the limit may exist even if \(f^{\prime \prime}(x)\) does not. \(\boldsymbol{H}\)

Let \(f\) be a differentiable real function defined in \((a, b)\). Prove that \(f\) is convex if and only if \(f^{\prime}\) is monotonically increasing. Assume next that \(f^{\prime \prime}(x)\) exists for every \(x \in(a, b)\), and prove that \(f\) is convex if and only if \(f^{\prime \prime}(x) \geq 0\) for all \(x \in(a, b)\).