An analytic function f(z)=z+a2z2 + … in the unit disk Δ = {z: |z| < 1} is said to be in $\mathcal{U}(\lambda, \mu)$ if ¶
$\left|f'(z)\left(\frac{z}{f(z)} \right)^ {\mu +1}-1 \right|\le \lambda \quad (|z|<1)$
¶ for some λ ≥ 0 and μ > –1. For –1 ≤ α ≤ 1, we introduce a geometrically motivated $\mathcal{S}_p(\alpha)$ -class defined by ¶
${\mathcal S}_p(\alpha) = \left \{f\in {\mathcal S}:\, \left |\frac{zf'(z)}{f(z)} -1\right |\leq {\rm Re}\, \frac{zf'(z)}{f(z)}-\alpha, \quad z\in \Delta \right \},$
¶ where ${\mathcal S}$ represents the class of all normalized univalent functions in Δ. In this paper, the authors determine necessary and sufficient coefficient conditions for certain class of functions to be in $\mathcal{S}_p(\alpha)$ . Also, radius properties are considered for $\mathcal{S}_p(\alpha)$ -class in the class ${\mathcal S}$ . In addition, we also find disks |z| < r:= r(λ,μ) for which $\frac{1}{r}f(rz)\in \mathcal{U(\lambda,\mu)}$ whenever $f\in \mathcal{S}$ . In addition to a number of new results, we also present several new sufficient conditions for f to be in the class $\mathcal{U}(\lambda, \mu)$ .