The Study of Operator monotone Functions by Analytic Continuation and Its Application

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We have almost achieved the purpose of this research project, that is, constructing new family of operator monotone functions with analytic continuation method. Actually we got the following- if a real monic polynomial has only real zeros, then the inverse function of the increasing part is operator monotone, and that if it has complex zeros, then the inverse function is semi-operator monotone. We have shown this result in the paper published from Transactions of American Mathematical Society We conjectured originally this problem and solved it with analytic continuation method. We hope this will be applied to orthogonal polynomials which are very important in many fields of mathematics.