Sharp bounds for the Neuman mean in terms of the quadratic and second Seiffert means

摘要

In this paper, we prove that α = 0 and β = 3 π ? 4 log ( 2 + 3 ) ( 2 π ? 4 ) log ( 2 + 3 ) = 0.29758 ? are the best possible constants such that the double inequality α Q ( a , b ) + ( 1 ? α ) T ( a , b ) 0 with a ≠ b , where Q ( a , b ) = ( a 2 + b 2 ) / 2 , S C A ( a , b ) = ( a ? b ) 3 ( a 2 + b 2 ) + 2 a b 2 ( a + b ) sinh ? 1 ( ( a ? b ) 3 ( a 2 + b 2 ) + 2 a b ( a + b ) 2 ) and T ( a , b ) = ( a ? b ) / [ 2 arctan ( ( a ? b ) / ( a + b ) ) ] are the quadratic, Neuman and second Seiffert means of a and b, respectively. MSC:26E60.