# Convexity conditions for $f$ and $\dfrac {1} {f}$

Let $f:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}$ be a real function.

Find all conditions on $f$ under which

$f$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$$\Leftrightarrow$ $\dfrac {1} {f}$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$

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Do you want us to solve the problem, or to help you with its solution? –  Ilya Aug 28 '12 at 8:01
By $\mathbb{R}_{+}$ you mean $(0,\infty)$? –  Nirakar Neo Aug 28 '12 at 8:04
@ Nirakar Neo : Yes. –  Maria Mikolayevskaya Aug 28 '12 at 8:11
@ Ilya: to solve the problem, if possible. –  Maria Mikolayevskaya Aug 28 '12 at 8:12
The question remains a bit unclear to me. Is the clause to be for a fixed interval $(a,b)$, i.e. so that the clause is not met if $f$ is convex in part of the interval and concave in another, or for any interval? Note that if $f$ is concave in any interval, $1/f$ must be convex in that interval. –  Einar Rødland Aug 28 '12 at 10:09