An inequality with positive numbers

If $a,b,c$ are positive and the sum of any two greater than the third, prove that ${\left(1+\frac{b-c}{a}\right)}^a{\left(1+\frac{c-a}{b}\right)}^b{\left(1+\frac{a-b}{c}\right)}^c<1$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Miscellaneous Examples, No. 293