*Correspondence: [email protected] 1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia Full list of author information is available at the end of the article Abstract The estimates for the second Hankel determinant a2a4 – a3 of the analytic function f (z) = z + a2z + a3z + · · · , for which either zf ′(z)/f (z) or 1 + zf ′′(z)/f ′(z) is subordinate to a certai...

Argument estimates for certain analytic functions associated with starlike functions of reciprocal order are obtained and special cases of the main results are indicated.

Ruscheweyh‘and She&Small proved the P6lya-Schoenberg conjecture that the class of convex analytic functions is closed under convolution or Hadamard product. They also showed that clos&o-convexity is preserved under convolution with convex analytic functions. In this note, we investigate harmonic analogs. Beginning with convex analytic functions, we form certain harmonic functions which preserve...

A parabolic starlike function f of order ρ in the unit disk is characterized by the fact that the quantity z f ′(z)/ f (z) lies in a given parabolic region in the right half-plane. Denote the class of such functions by PS∗(ρ). This class is contained in the larger class of starlike functions of order ρ. Subordination results for PS∗(ρ) are established, which yield sharp growth, covering, and di...

Let ∑ p be the class of functions f(z) which are analytic in the punctured disk E∗ = {z ∈ C : 0 < |z| < 1}. Applying the linear operator Dn+p defined by using the convolutions, the subclass n+p(α) of ∑ p is considered. The object of the present paper is to prove that n+p(α) ⊂ n+p−1(α). Since 0(α) is the class of meromorphic p-valent starlike functions of order α, all functions in n+p−1(α) are m...