Abstract
We establish two optimal inequalities among power meanMp(a,b)=(ap/2+bp/2)1/p, arithmetic meanA(a,b)=(a+b)/2, logarithmic meanL(a,b)=(a−b)/(loga−logb), and geometric meanG(a,b)=ab.
Highlights
For p ∈ R, the power mean Mp a, b of order p of two positive numbers a and b is defined by ⎧ Mp a, b ⎪⎨ ap bp ⎪⎩√ab,2 1/p, p / 0, p 0.In the recent past, the power mean Mp a, b has been the subject of intensive research
The power mean Mp a, b has been the subject of intensive research
Many remarkable inequalities for the mean can be found in literature 1–11
Summary
In 12 , Alzer and Janous established the following sharp double inequality see 1.5 for all real numbers a, b > 0, and the constant 1/3 in the left side inequality cannot be improved. The main purpose of this paper is to present the optimal bounds for Aα a, b L1−α a, b and Gα a, b L1−α a, b for all α ∈ 0, 1 in terms of the power mean Mp a, b .
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
