Initial coefficients and fourth Hankel determinant for certain analytic functions

Abstract

The present work is an attempt to give partial proofs of certain conjectures on the fifth coefficient of certain normalized analytic functions. Further, bounds on the sixth and seventh coefficients for the starlike functions related to a lune are also investigated. The non-sharp bound on third and fourth Hankel determinants are also obtained.

Sharp bounds of third Hankel determinant for a class of starlike functions and a subclass of $q$-starlike functions

“…On differentiating h 2 with respect to p, we obtain ∂h 2 ∂p = 6p 5 (1 + q) 3 (−1 + 7q + 19q 2 + 9q 3 + q 4 + q 5 ) + 2048p(77 + 146q + 246q 2 + 140q 3 + 94q 4 + 6q 5 + 3q 6 ) − 128p 3 (929 + 1783q + 2636q 2 + 1666q 3 + 878q 4 − 4q 5 + 29q 6 + 3q 7 ), further which becomes 0 at p = 0 and p = p 0 , given by p 0 := 32(929 + 1783q + 2636q 2 + 1666q 3 + 878q 4 − 4q 5 + 29q 6 + 3q 7 ) 3(1 + q) 3 (−1 + 7q + 19q 2 + 9q 3 + q 4 + q 5 ) − Ã, where à = 64 √ 2A 0 3(−1 + 4q + 37q 2 + 86q 3 + 92q 4 + 50q 5 + 15q 6 + 4q 7 + q 8 ) and A 0 = (107909 + 414041q + 1008402q 2 + 1557100q 3 + 1804144q 4 + 1471838q 5 + 913014q 6 + 363176q 7 + 107408q 8 + 9900q 9 + 6570q 10 + 346q 11 + 41q 12 + 15q 13 ) (1/2) .…”

“…Also, another type of second order Hankel determinant is obtained by taking q = 2 and n = 3, mathematically written as H 2 (3) := a 3 a 5 − a 2 4 . The estimations of the sharp bounds for these H q(n) is obtained by many authors for various sub-classes of A (see [5,23,29]). Third order Hankel determinant, given by H 3 (1) = a 3 (a 2 a 4 − a 2 3 ) − a 4 (a 4 − a 2 a 3 ) + a 5 (a 3 − a 2 2 ), (1.1) is obtained when q = 3 and n = 1.…”

Third Hankel determinant for q -analogue of symmetric starlike connected to q -exponential function

“…4 . The estimations of the sharp bound for these H q (n) is obtained by many authors for various sub-classes of A ( [3,23,33]). If we set q = 3 and n = 1 we get the |H 3 (1)| , the third order Hankel deerminant, which is given by…”

"Fourth Hankel determinant for a subclass of analytic functions defined by generalized Sălăgean operator"

“…The initiative of finding the fourth Hankel determinant for subclasses of analytic functions was taken by Arif et al [3] in 2018. After that, only a few reserachers worked in this direction including [32,4,13,33,10,31].…”