UNIFORMLY Starlike UNIVALENT functionS INTRODUCED BY GOODMAN AND WE DEVELOP THIS IDEA OVERHARMONIC functionS. WE INTRODUCE A SUBCLASS OF HARMONIC UNIVALENT functionS WHICH ARE FULLYStarlike AND UNIFORMLY Starlike ALSO. IN THE FOLLOWING WE WILL MENTION SOME EXAMPLES OF THIS SUBCLASS AND OBTAIN TWO NECESSARY AND SUFFICIENT CONDITIONS, ONE WITH THE INEQUALITY FORM ANDOTHER WITH CONVOLUTION.

LET P (G,B), G>0, B <1 DENOTE THE CLASS OF ANALYTIC function F IN THE UNIT DISK NORMALIZED BY F (0) =1, F′ (0)=1 AND SATISFYING THE CONDITION (FORMULA) FOR SOME J Î R. IN THIS PAPER CONSIDER SN(A), THE CLASS OF N-Starlike function OF ORDER , DEFINED BY G. S. SALAGEAN (1983) [5] AND WE FIND CONDITION ON G, B SO THAT P(G,B)ÍSN (A). WE TAKE ADVANTAGE OF THE RUSCHEWEH’S DUALITY THEORY.

IN THE PRESENT PAPER, A NEW SUBCLASSES OF A-Starlike functionS WITH RESPECT TO (J, K) –SYMMETRIC POINTS WITH NEGATIVE COEFFICIENTS AND A-CONVEX functionS WITH RESPECT TO (J, K) -SYMMETRIC POINTS WITH NEGATIVE COEFFICIENTS WAS DEFINED AND INVESTIGATED. FURTHER, THE COEFFICIENTS INEQUALITY FOR THE function IN THIS CLASSES WAS OBTAINED. IN WHAT FOLLOWS, SOME TEST EXAMPLE WAS OBTAINED.

IN THIS PAPER WE INTRODUCE AND INVESTIGATE A CERTAIN SUBCLASS OF UNIVALENT functionS WHICH ARE ANALYTIC IN THE UNIT DISK U AND STATE SUCH RESULTS AS COEFFICIENT INEQUALITIES. THE RESULTS PRESENTED HERE WOULD PROVIDE EXTENSIONS OF THOSE GIVEN IN EARLIER WORKS.

IN THIS PAPER, WE FINDING CONDITIONS ON THE TRIPLET (A, B, C) SO THAT THE function ZF(A, B; C; Z) IS Starlike IN D, WHERE F (A, B; C; Z) DENOTES THE HYPER GEOMETRIC function.ALSO THE GEOMETRIC PROBLEM OF STAR LIKENESS AND CLOSE TO CONVEXITY OF ZF (A, B; C; Z) IS STUDIED.