The Univalence And Inclusion Relations Of The Function Race

Content: In this thesis, we has investigated the problems of some criteria of univalent functions and relations between the convolution and differential operators. Also, we has studied the inclusion relations to the class of analytic functions and harmonic univalent functions with negative coefficients . The second chapter, using convolution operator, we obtain some criteria for functions to become univalent, and had proven the criteria with the convolution operator is broader than the differentialoperator. The third chapter, a new class R(m,λ, μ,α) introduced by Liu mingsheng isdiscussed. We had made the reasonable promotion and improvement to this class and relative inclusion relations. At the same time, we obtain a new promoted function raceC_n(m,λ, μ,α). What is more, we show the sufficient conditions for the functions belong to C_n(m,λ, μ,α) to become univalent. The fourth chapter, We has introduced a class HS(β) of harmonic univalent functions with negative coefficients, anddiscussed the subclass HS~*(β) of this function race the sufficient and necessaryconditions, distortion theorems and so on.