Let A denote the class of funetions f(z) defined by f(z)=z-sum from n=2 to ∞a_nz^n (a_n>0) (1) which are analytic in the unit disk U= {z:|z|<1}. A function f(z)∈A is said to be in the class S (x,β,γ), if and only if |(f′(z)-1)/(af′(z)+1-(1+α)β)|<γ(2) for 0
Let P(Z) = 1 + P1Z + P2Z2 +…be an analytic function in the unit disc D. In thispaper, we determine the value of φ(α,β) for whichRe[P(Z) + αZP’(Z)] > β,Z ∈ D,α > 0,β < 1implies that Re{P(Z)} > <φ(α,β) for all Z ∈ D, and this result is sharp. Some of its interesting consequences are also given. In addition, we give a new univalence criterion.
