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    On Relation between Analytic and Univalent Functions Defined by Close-to P Class with the Function Belonging to S Class
    (Amer Inst Physics, 2017) Yıldız, İsmet; Uyanık, Neslihan; Albayrak, Hilal; Ay, Hilal
    The Weierstrass's associated function is not elliptic but it is of great use in developing the theory of elliptic function. The Zeta function is defined by the double series Sigma(m)'Sigma(m)''{1/z-W-mn + 1/W-mn + z/W-mn(2)}, where W-mn = 2m omega(1) + 2n omega(2) and m, n are integers, not simultaneously zero; the summation Sigma(m)'Sigma(m)''{1/z-W-mn + 1/W-mn + z/W-mn(2)} extends overall integers, not simultaneously. Which W-mn are Lattice points. Evidently W-mn are simple poles of zeta(z) and hence the function is meromorphic in W = {m omega(1) + n omega(2) : (m, n) not equal (0, 0), m, n is an element of Z, Im tau > 0}, D* = {z : vertical bar z vertical bar > 1, vertical bar Rez vertical bar < 1/2 and Im tau > 0, z is an element of C}. zeta(z) is uniformly convergent series of analytic functions, so the series can be differentiated term-by-term. zeta(z) is an odd function, hence the coefficients of the terms z(2k) is evidently zero when k is positive integers. Let A be the class of functions f (z) which are analytic and normalized with f (0) = 0 and f' (0) = 1. Let S be the subclass of A consisting of functions f (z) which are univalent in D. Let P class be univalent functions largely concerned with the family S of functions f analytic and univalent in the unit disk D, and satisfying the conditions f (0) = 0 and f'(0) = 1. One of the basic results of the theory is growth theorem, which asserts in part that for each f is an element of S. In particular, the functions f is an element of S are uniformly bounded on each compact subset of D. Thus the family S is locally bounded, and so by Montel's theorem it is a normal family. A relation was established between S class with function of Weierstrass which is analytic and monomorphic Closes-to-P class in unit disk.