Show that $1-x^{\frac{1}{1-t}}+\frac{1}{1-t}x^{\frac{1}{1-t}}\ln x$ is positive over domain $x\in(0,1)$

Question

I would like to demonstrate that the following function is positive \begin{equation*} 1-x^{\frac{1}{1-t}}+\frac{1}{1-t}x^{\frac{1}{1-t}}\ln(x), \end{equation*} where $1>x>0$ and $1>t>0$.

I have graphed this in Desmos and confirmed that the function is positive given my restrictions on $x$ and $t$.

Any help to solve this analytically would be most appreciated!

Many thanks.

Answer 1

Let $a=x^{\frac{1}{1-t}}$ and rewrite the expression as a function of $a$,

$$1-x^{\frac{1}{1-t}}+\frac{1}{1-t}x^{\frac{1}{1-t}}\ln x=1-a+a\ln a =f(a)$$

Since $1>x>0$ and $1>t>0$, the domain for $a$ is $0<a<1$.

Evaluate the derivative over its domain

$$f'(a) = \ln a < 0$$

that is $f(a)$ is strictly decreasing over $0<a<1$. Thus,

$$f(a) > f(1) =0$$