Double integral operators which were considered by S. S. Miller and P. T. Mocanu Integral Transform. Spec. Funct. 19 2008 , 591-597 are discussed. In order to show the analytic function f z is starlike of order β in the open unit disk U, the theory of differential subordinations for analytic functions is applied. The object of the present paper is to discuss some interesting conditions for f z to be starlike of order β in U concerned with second-order differential inequalities and double integral operators.

Let𝒜pbe the class of certain analytic functionsf(z)in the open unit diskU. Forf(z)∈𝒜p, a subclass𝒜p(α,β,γ,j)of𝒜pis introduced. The object of the present paper is to discuss some properties of functionsf(z)belonging to classAp(α,β,γ;j).

Let f (z) be an analytic function in the open unit disk D normalized with f (0) = 0 and f (0) = 1. With the help of subordinations, for convex functions f (z) in D, the order of close-to-convexity for f (z) is discussed with some example. MSC: Primary 30C45

Let $\mathcal{A}$ be the class of functions $f(z)$ which are analytic in the open unit disk $\mathbb{U}$ with $f(0)=0$ and $f'(0)=1$. For the class $\mathcal{A}$, a new general class $\mathcal{A}_{k}$ is defined. With this general class $\mathcal{A}_{k}$, two interesting classes $\mathcal{S}_{k}^{\ast}(\alpha)$ and $\mathcal{K}_{k}(\alpha)$ concerning classes of starlike of order $\alpha$ in $\mathbb{U}$ and convex of order $\alpha$ in $\mathbb{U}$ are considered.

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