Recently, i read the following paper: "Smail DJEBALI - Uniform continuity and growth of real continuous functions, International Journal Mathematical Education in Science and Technology, Vol. 32, No 5(2001) 677–689"
In the paper, he noticed the following result as an appendix:
Let $~f:I=[a,\infty)\to\mathbb{R}$ be a continuous function defined by $f(x)=x^{\alpha}\sin(x^{\beta})$, where $a>1$. Then, $~f$ is continuous uniformly on $I$ if and only if a pair $(\alpha,\beta)\in\mathbb{R}^{2}$ is an element of the following set: $$\{(\alpha,\beta)~|~\alpha<0\}\cup\{(\alpha,\beta)~|~\alpha+\beta\le1\}.$$
I wonder more general case of the above result, such as
(Q1) Let $~f:I=(0,\infty)\to\mathbb{R}$ be a continuous function defined by $f(x)=x^{\alpha}\sin(x^{\beta})$.
Find all pairs of $(\alpha,\beta)$ for which $~f$ is continuous uniformly on $(0,\infty)$.
(Q2) Let $~f:I=(0,\infty)\to\mathbb{R}$ be a continuous function defined by $f(x)=x^{\alpha}\cos(x^{\beta})$.
Find all pairs of $(\alpha,\beta)$ for which $~f$ is continuous uniformly on $(0,\infty)$.
Possible answer for (Q1) is (I'm not ensure about it): $$\{(\alpha,\beta)~|~\alpha=0,\,0\le\beta\le1\}\cup\{(\alpha,\beta)~|~-\beta\le\alpha<0\}\cup\{(\alpha,\beta)~|~0<\alpha\le1-\beta\}.$$
Deducing the above answer, inequalities for sine function are useful to me.
But, in the case of (Q2), there is no useful inequality for a cosine function like a sine function, so I've not got a possible answer.
Any help or Give some advice! Thank you!