A coefficient inequality for a subclass of the Carathéodory functions defined using conical domains

For 0≤k<∞0≤k<∞, let ΩkΩk be the conical domain in the complex plane CC defined by Ωk={w∈C:w=u+iv,u2>k2((u−1)2+v2),u>0}. Let qk(z)qk(z) be the Riemann map of U:={z∈C:|z|<1}U:={z∈C:|z|<1} onto ΩkΩk satisfying qk(0)=1,qk′(0)>0. Let P(qk)P(qk) be the class of analytic functions h(z)h(z) subordinate in UU to qk(z)qk(z) and represented by h(z)=1+b1z+b2z2+⋯h(z)=1+b1z+b2z2+⋯, (z∈U)(z∈U). Sharp estimates for |b2−ub12|(−∞