Bounded Positive Function Derivative And The Coefficient Estimates

This dissection is mainly to discuss the problems of estimating in the derivation of bounded functions, the coefficients of bounded regular functions and Schwarz-Pick Lemma for the derivation on the hyperbolic metric.In first chapter, we easily introduce status in quo of the problem of the derivative of bounded regular functions and the problem of coefficients estimating of bounded regular functions, and some results is obtained in this paper.In second chapter, we discuss the problems of estimating in the derivation of bounded functions. A general estimation of the nth derivation of the function is presented by using the principle of the inductive model and the characters of the bounded functions. Some results of bounded regular functions have been given from special to general. Hence, there are some better results in the problem of estimating the derivation of bounded regular functions.In third chapter,the problem of estimating coefficients of bounded regular functions (i.e. the problem of Landau) is discussed. We present the coefficient series of Taylor series by { an}, and sup{a0 + a1 + a2 + a3} is obtained, and on univalentbounded regular functions sup{a0 +a1 + a2} is gotten. Also a result of Zdzislaw,Lewandowski and Jan Szynal is generalized.In forth chapter, we have discussed Schwarz -Pick Inequality on the the derivative of hyperbolic metric. We have generalized Schwarz-Pick Lemma from Schwarz -Pick Lemma of the derivation and Schwarz -Pick Lemma on the hyperbolic metric to Schwarz -Pick inequality of the derivation on the hyperbolicmetric, A result of A.F.Beardon and Peter R.Mercre is generalized to obtain the Schwarz-Pick Lemma for the hyperbolic derivation.