The main aim of this paper is to discuss the third Hankel determinants for three classes: \(S^*\) of starlike functions, \(\mathcal {K}\) of convex functions and \(\mathcal {R}\) of functions whose derivative has a positive real part. Moreover, the sharp results for twofold and threefold symmetric functions from these classes are obtained.

Project Euclid - mathematics and statistics online

We consider the Fekete-Szegö inequalities for classes which were defined by Murugusundaramoorthy et al. (2013). These inequalities will result in bounds of the third coefficient which are better than these obtained by Murugusundaramoorthy et al. (2013). Moreover, we discuss two other classes of bi-univalent functions. The estimates of initial coefficients in these classes are obtained.

So far, the sharp bound of the expression |a2a4−a32| for the class C of close-to-convex functions has remained unknown. In this paper, we obtain the estimation of this expression, called the second Hankel determinant, for C0, i.e. the subset of C consisting of functions f that satisfy in the unit disk the inequality Re(zf′(z)/g(z))>0 with a starlike function g.Moreover, some remarks on the second Hankel determinant for the class S of univalent functions are made. It is proven that max⁡{|a2a4−a32|:f∈S} is greater than 1.

In this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$S S ∗ and $${\mathcal {K}}_S$$K S of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$S S ∗ and $${\mathcal {K}}_S$$K S by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$S S ∗ and $${\mathcal {K}}_S$$K S .