Study on the properties for partial sums of analytic functions

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Let f(z) be analytic in the open unit disk. By using differential operators, integral operators, convolutions, subordinations and differential equations, we study some properties of analytic functions and rational functions.In general, partial sums fn(z) of starlike functions and convex functions don't keep their Properties of functions f(z). Paper : 『S.Owa, Partial sums of certain analytic functions, International Journal of Mathematics and Mathematical Sciences Vol.25, No.12(2001), 771-775』 shows the radius of starlikeness or convexity of partial sums of extremal functions for classes of starlike functions or convex functions. More recently, we can see one idea for discussing such problem for partial sums. Furthermore, we are now trying to consider the partial sums of analytic functions of several complex variables.For the fractional calculus operators, we consider the paper : 『J.Littlewood, On inequallities in the theoriy of functions, Proceedings of the London Mathematical Society Vol.23(1925), 481 ・・ 519』 Considering this paper, we know that we can discuss partial sums concern with the fractional calculus operators of analytic functions. Therefore, we would like to study partial sums of analytic functions with such new ideas and problems which we have during this study between 1999 and 2001 after finishing this project.