A new family of operator means is introduced. It interpolates the arithmetic, geometric, harmonic and logarithmic means. Moreover it includes some special operator means, for example, the power difference, Stolarsky and identric means, continuously. Then order relations among them are obtained.

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We consider operator monotonicity of a 2-parameter family of functions including the representing function of the Stolarsky mean, which is constructed by integration of the function [(1 − α) + αx p ] 1 p , representing the weighted power mean, of α ∈ [0,1]. We also think about operator monotonicity of exp{ f (x)} for a continuous function f (x) defined on (0,∞) .

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We obtain an integral representation of holomorphic function P α (z) which is real on the positive part of the real axis and formedFor this purpose we define a two variable function which is substituted for an argument θ , and also find an explicit real and imaginary part of P α (x + iy) .