Generalized Zalcman conjecture for some classes of analytic functions

For functions f(z)=z+a2z2+a3z3+⋯ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional ϕ(f,n,m;λ):=|λanam−an+m−1|. For all real parameters λ and β<1, we provide the sharp upper bound of ϕ(f,n,m;λ) for functions f satisfying Ref′(z)>β and hence settle the open problem of estimating ϕ(f,n,m;λ) recently proposed by Agrawal and Sahoo (2016) [1]. For all real values of λ, the estimations of ϕ(f,n,m;λ) are provided for starlike and convex functions of order α (α<1) which are sharp for λ≤0 or for certain positive values of λ. Moreover, for certain positive λ, the sharp estimation of ϕ(f,n,m;λ) is given when f is a typically real function or a univalent function with real coefficients or is in some subclasses of close-to-convex functions.