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It makes me think of means... The RHS Is like a geometric mean but the "divided by 4" annoyes me. The LHS is probably an arithmetic mean... Can they be combined?

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By Cauchy-Schwarz, $$\sqrt{\frac{a^2 + b^2 + c^2 + d^2}{4}} =\sqrt{a^2 + b^2 + c^2 + d^2}\cdot \sqrt{4\cdot\frac{1}{4^2}}\geq \frac{a+b+c+d}{4}\\\geq \sqrt[3]{\frac{abc + abd + acd + bcd}{4}}$$ where in the last step we use Inequality. $\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab$

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Robert Z
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    The first inequality $\sqrt{\frac{a^2 + b^2 + c^2 + d^2}{4}} \geq \frac{a+b+c+d}{4}$, in addition to being a special case of CS, actually has its own name: The quadratic mean-arithmetic mean inequality. – Arthur Sep 03 '16 at 12:53