Question Number 154596 by mathdanisur last updated on 19/Sep/21
$$\mathrm{let}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{z}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }}\:+\:\sqrt{\mathrm{2}}\:\geqslant\:\mathrm{2}\:\sqrt{\frac{\mathrm{z}}{\mathrm{x}+\mathrm{y}}}\:+\:\frac{\sqrt{\mathrm{2xy}}}{\mathrm{x}+\mathrm{y}} \\ $$