GENERALIZED POSITIVITY TECHNIQUES APPLIED TO HYPERGEOMETRIC TYPE FUNCTIONS

Abstract

Positivity of trigonometric sums has always been an important area of research because of its versatility of applications. The objective of the present thesis is to study the positivity of trigonometric polynomials for a particular type of non-negative sequence, which is generalization of the sequences studied earlier in the literature. We are interested in nding the applications of positivity of trigonometric sums in geometric function theory because using this technique we get better results than the results available in speci c cases of geometric function theory. We apply the positivity of trigonometric sums to hypergeometric type functions in particularly to the Ces aro mean of type (b 􀀀 1; c) to study its geometric behaviour. In this work, for fakg be a sequence of positive real numbers, we study positivity of trigonometric sums Xn k=0 ak cos k and Xn k=1 ak sin k for 2 (0; ) and n 2 N. We choose a new sequence ak = qk such that q0 = 2; q1 = 1; qk = 1 (k + ) (k + ) ; k 2; where ; 0. The conditions on ; ; and are obtained such that the corresponding sine and cosine sums are positive. Furthermore, the conditions on and are modi ed to < 0 and > 0 and careful analysis resulted in concluding that the positivity of sine sum can be retained whereas a similar conclusion could not be drawn for the cosine sums, which requires further analysis. The obtained results are extended to analyse the geometric properties of Ces aro mean of type (b 􀀀 1; c) which is the generalization of Ces aro mean of order for particular case b = 1 + and c = 1. Further we choose another sequence ak = ck such that c2k = c2k+1 = Bn􀀀k Bn (1 􀀀 )k k! ; k = 0; 1; 2; : : : ; n; i where B0 = 1 and Bk = (b)k (c)k 1+b􀀀c b . We determine the range on so that simultaneously the positivity of sine and cosine sums holds. These new results are applicable in nding the location of zeros of trigonometric polynomials and the positivity of a new orthogonal polynomial sum. The interpretation of positivity of sine and cosine sums for fckg is re ected in terms of generalized Ces aro stable function. Moreover, the results obtained give slightly better approximation in the sense of subordination. This concept of generalized Ces aro stable function can also be retrieved via lower triangular matrices. Further two conjectures are also posed in the direction of generalized Ces aro stable function. Consequences involving Gegenbauer polynomials are also deduced in the context of Kakeya Enestr om theorem which gives the conditions on the coe cients of a polynomial so that the approximants are zero-free in D. Positivity of sine and cosine sums for fckg also provide results related to the starlikenenss of zF(a; b; c; z) for certain ranges of a, b and c. Comparison of obtained results with the analogue results (which are obtained using di erent coe cient sequence already in the literature) are also provided. Further, conditions on the coe cients of a polynomial are also obtained to nd the radius such that the polynomial in discussion is convex in D := fz : jzj < g. For positivity of trigonometric sums in correlation with starlike functions, Koumandos and Ruscheweyh posed a conjecture which evolved from the concept of stable functions. Already the conjecture is establish for few values. In this thesis, we establish the conjecture for all in neighbourhood of 1=3 and in the weaker form for = 2=3. It is expected that for the same values the conjectures proposed in the general setup can be validated. Several consequences of obtained results related to starlike functions and particular orthogonal polynomials sums including Gegenbauer polynomials are also outlined. Open problems and future scope are mentioned wherever possible.