MathDB

For

p

q

l

strictly positive real numbers, consider the following problem: for

y \geq 0

fixed, determine the values

x \geq 0

such that

x^p - lx^q \leq y

. Denote by

S(y)

the set of solutions of this problem. Prove that if one has

p < q

\epsilon \in (0, l^\frac{1}{p-q})

0 \leq x \leq \epsilon

and

x \in S(y)

, then

x\leq ky^{\delta},\ \text{where}\ k=\epsilon\left(\epsilon^p-l\epsilon^q\right)^{-\frac{1}{p}}\ \text{and}\ \delta=\frac{1}{p}

Problem Statement