MathDB

Let

a

and

b

be positive numbers. Find the largest number

c

, in terms of

a

and

b

, such that for all

x

with

0<|x|\le c

and for all

\alpha

with

0<\alpha<1

, we have:

a^\alpha b^{1-\alpha}\le\frac{a\sinh\alpha x}{\sinh x}+\frac{b\sinh x(1-\alpha)}{\sinh x}.

B6