ineqalities.............................

A.M of nth powers > nth power of A.M
\left(\frac{a^{n}+c^{n}}{2} \right)>\left(\frac{a+c}{2} \right)^{n}\Rightarrow a^{n}+c^{n}>2\left(\frac{a+c}{2} \right)^{n} ....................(1)

but we know AM >GM>HM
so AM > HM
a+b2 > b
= \left( \frac{a+c}{2}\right)^{n}>b^{n}
and
= 2\left( \frac{a+c}{2}\right)^{n}>2b^{n} ......................(2)
from (1) and (2)
we get
a^{n}+c^{n}>2b^{n} [ Proved ]

a+c=2ac/b
a^n+c^n>2(a/2+c/2)^n>2(ac/b)^n (mean power)

now a.m≥g.m.1/b≥(1/ac)^.5
so,ac>b^2≡ac/b>b,,,,,putting this in 1st equation.we get what we need