A Note on Schur-Convexity of Extended Mean Values

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The Schur Geometrical Convexity of the Extended Mean Values (2008, unknown): “Qi [19] first obtained the following result.”
Bounds for the Ratio of Two Gamma Functions (2009, methodology): “) 6= 0; E(0,0;x,y) = √ xy, x6= y; E(r,s;x,x) = x, x= y. (3.7) Actually, the inequality (3.6) is an immediate consequence of monotonicity of E(r,s;x,y), see [30]. For more information, please refer to [13, 17, 23, 33, 38, 39, 47, 48, 50, 51, 58, 70, 71, 73, 74, 75, 78] and related references therein.”
An alternative note on the Schur-convexity of the extended mean values (2006, background): “Theorem A ([11]).”
The Best Bounds in Gautschi-Kershaw Inequalities (2006, unknown): “The positive function g a , b ( u ) has been researched in [ 18 ] and was applied in [ 7, 10, 12, 17 ] to prove the logarithmic convexity or the Schur convexity of the extended mean values.”
Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave (2008, background): “The Schur-convexity of the extended mean values E(r, s; x, y) with respect to (r, s) and (x, y) are investigated in [9], [10], and [11].”