Generalization of partial sums of certain analytic and univalent functions

Let ϕ(z)ϕ(z) be a fixed analytic and univalent function of the form ϕ(z)=z+∑k=2∞ckzk and Hϕ(ck,δ)Hϕ(ck,δ) be the subclass consisting of analytic and univalent functions f(z)f(z) which satisfy the inequality ∑k=2∞ck|ak|<δ. In this paper, we study the ratio of a function of the form f(z)=z+∑k=2∞akzk to its sequence of partial sums of the form fn(z)=z+∑k=2nakzk where the coefficients of f(z)f(z) satisfy the above condition. Also, we determine sharp lower bounds for Re{f(z)/fn(z)}, Re{fn(z)/f(z)}, Re{f′(z)/fn′(z)} and Re{fn′(z)/f′(z)}.