On some classes of analytic functions (Q5906911)

The authors study generalisations of close-to-convex functions, defined as follows: 1) For \(b\in\mathbb{C} \setminus \{0\}\), \(h\in P(b)\) if and only if \((h-1+b)/b\) is a function with positive real part normalized as usual. 2) For \(k\geq 2\), \(b\) as above, \(H\in P_k(b)\) if and only if there exist \(h_1\) and \(h_1\in P_k\) such that \[ H(z)= \left({k\over 4} +{1\over 2} \right) h_1(z)- \left({k \over 4} -{1\over 2} \right) h_2(z),\;| z|<1. \] 3) For \(k,b\) as above, \(f\in K_k(b)\) if and only if there exists a convex function \(g\) such that \(f'/g' \in P_k(b)\). For such functions \(f\) they derive results on Hadamard convolution, inclusion relations and theorems on the Taylor coefficients and linear combinations.