Carleman’s inequality

Theorem ([1], pp. 24)For positive real numbers ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$,Carleman’s inequality states that

$$\sum _{n=1}^{\mathrm{\infty }}{\left(a_1{a}_2\mathrm{\cdots }{a}_n\right)}^{1/n}\le e\sum _{n=1}^{\mathrm{\infty }}{a}_n.$$

Although the constant $e$ (the natural log base)is optimal, it is possible to refineCarleman’s inequality by decreasing the weight coefficientson the right hand side [2].