The Fekete-Szego coefficient functional problems for q-starlike and q-convex functions related with lemniscate of Bernoulli

The area of q-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of q-calculus was initiated by Jackson [Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193-203; On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253-281.], who was the first to develop q-integral and q-derivative in a systematic way. In this paper, we make use of the concept of q-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of q-starlike and q-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this q-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of vertical bar a(3) mu a(2)(2)vertical bar found in [Mishra and Gochhayat, Fekete-Szego problem for k-uniformly convex functions and for a class defined by Owa-Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563-572] for f is an element of k-UCV, 0 <= k < infinity.