Abstract
We find the least values p, q, and s in (0, 1/2) such that the inequalities H(pa + (1 − p)b, pb + (1 − p)a) > AG(a, b), G(qa + (1 − q)b, qb + (1 − q)a) > AG(a, b), and L(sa + (1 − s)b, sb + (1 − s)a)> AG(a, b) hold for all a, b > 0 with a ≠ b, respectively. Here AG(a, b), H(a, b), G(a, b), and L(a, b) denote the arithmetic‐geometric, harmonic, geometric, and logarithmic means of two positive numbers a and b, respectively.
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