Bounds for the surface area of an ellipsoid

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Problem 2.94.2 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 2.94 "[NO TITLE]"

Let EE denote the area of the surface of an ellipsoid with semi-axes a,b,ca, b, c and prove that

4π(bc+ca+ab)3≤E≤4π(a2+b2+c2)3. \frac{4\pi (bc+ca+ab)}{3} \leq E \leq \frac{4\pi (a^{2}+b^{2}+c^{2})}{3} .
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Remark.
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Derive, or take for granted, that

E=∬(b2c2ξ2+c2a2η2+a2b2ζ2)12 dω; E = \iint (b^{2}c^{2}\xi ^{2} + c^{2}a^{2}\eta ^{2} + a^{2}b^{2}\zeta ^{2})^{\frac{1}{2}} \, d\omega ;

the integration is extended over the surface of the unit sphere of which (ξ,η,ζ)(\xi , \eta , \zeta ) is a point,

ξ2+η2+ζ2=1, \xi ^{2} + \eta ^{2} + \zeta ^{2} = 1,

and dωd\omega the surface element,

∬dω=4π. \iint d\omega = 4\pi .
Problem 2.94.3 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 2.94 "[NO TITLE]"

(continued). The surface area of the ellipsoid is larger than the surface area of the sphere with the same volume, that is

E>4π(abc)23 E > 4\pi (abc)^{\frac{2}{3}}

unless a=b=ca = b = c.

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References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.