Determine the largest interval $]\alpha,+\infty[$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{\sqrt{a^2+b^2}}<1$ for every $a,b \in ]\alpha,+\infty[$?

Question

Let $a$ and $b$ be two (strictly) positive real numbers (i.e., $a,b \in ]0,+\infty[$).

How to determine the largest interval $]\alpha,+\infty[$ (that is contained in $]0,+\infty[$) such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{\sqrt{a^2+b^2}}<1$$ for every $a,b \in ]\alpha,+\infty[$?

Obviously, $]\alpha,+\infty[$ is contained in $]2,+\infty[$.

Answer 1

Assume $a=b=x$ such that

$$\frac{1}{a}+\frac{1}{b}+\frac{1}{\sqrt{a^2+b^2}}=\frac 2 x+\frac{1}{\sqrt{2x^2}}=1 \iff 2+\frac1{\sqrt 2}=x$$

Now observe that

$$f(a,b)=\frac{1}{a}+\frac{1}{b}+\frac{1}{\sqrt{a^2+b^2}}$$

is decresing with $a$ and $b$ therefore $\alpha=2+\frac 1 {\sqrt 2}$.