How can show $\left(1+\frac{1}{a}\right)\left(1-\frac{1}{\sqrt{2}}\int_{0}^{1}\sqrt{1+u^a}\,du\right)<1-1/\sqrt{2}$

Question

I was working on a problem and reduced it to showing the following inequality: ‎‎ $$\left(1+\frac{1}{a}\right)\left(1-\frac{1}{\sqrt{2}}\int_{0}^{1}\sqrt{1+u^a}\,du\right)<1-1/\sqrt{2};\quad a>0.$$ Your suggestion?

Answer 1

hint:

$\sqrt{\dfrac{x+y}{2}}\ge \dfrac{\sqrt{x}+\sqrt{y}}{2}$

but it is for $<\dfrac{1}{2}$

edit 1:

$\sqrt{\dfrac{1+u^a}{2}}>\dfrac{1+u^{\frac{a}{2}}}{2}$

now you can integral this function and go further which shouldn't be difficult .

edit 2:

lemma: for $0 \le x \le 1, \sqrt{1+x} \ge 1+(\sqrt2-1)x$

square both sides we have $(\sqrt2-1)x(1-x) \ge 0$ which is true.

so we have $\sqrt{1+u^a} \ge 1+(\sqrt2-1)u^a$

then you get the better result.