Coefficient Estimates for Certain Classes of Bi-Univalent Functions

A function analytic in the open unit disk <svg style="vertical-align:-0.0pt;width:14.6625px;" id="M1" height="11.175" version="1.1" viewBox="0 0 14.6625 11.175" width="14.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D53B" d="M810 350q0 -173 -123 -267q-107 -83 -297 -83h-352v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h405q194 0 282 -85q85 -82 85 -215zM712 336q0 80 -30 143t-88 97q-66 41 -170 41q-62 0 -78 -13q-13 -9 -13 -57v-416q0 -55 24.5 -75t81.5 -20q133 0 203 86 t70 214zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5t7.5 79.5z" /></g> </svg> is said to be bi-univalent in <svg style="vertical-align:-0.0pt;width:14.6625px;" id="M2" height="11.175" version="1.1" viewBox="0 0 14.6625 11.175" width="14.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D53B"/></g> </svg> if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in <svg style="vertical-align:-0.0pt;width:14.6625px;" id="M3" height="11.175" version="1.1" viewBox="0 0 14.6625 11.175" width="14.6625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D53B"/></g> </svg> makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions.