I have been trying to prove that the function \begin{equation} (2-x)^{2k-p}\int_0^1t^{\frac{k}{p}-1}(1-tx)^{-\frac{k}{p}}dt, \,\,\, x\in [0,1] \end{equation} attains its maximum at $x=1$ under the hypothesis that $k \geq 2, \,\,\, k+\sqrt{k(k-1)} \leq p \leq k+\sqrt{k(k-\frac{1}{2})}$.
Some plotting seems to suggest that the function is convex which would solve the problem, but I was not able to prove convexity analytically.
