Let A(p) denote the class of analytic functions f which are of the form \(f(z)=z^ p+..\). in the unit disk. Let B(p,\(\beta\),\(\gamma)\) with \(\beta >0\) and \(0\leq \gamma <p\) be its subclass which consists of functions f satisfying \(Re(zf'(z)/(f(z)^{1-\beta}g(z)^{\beta}))>\gamma\) where \(g\in A(p)\) is p-valently starlike. The main result of the present paper states that if \(f\in A(p)\) satisfies \[ Re((1-\alpha)(zf'(z)/f(z))+\alpha (1+(zf''(z)/f'(z))))>\beta, \] with \(\alpha\neq 0\), \(p<\beta\) and \(| \beta /\alpha | \leq\), then f is p-valent, and that, if moreover \(0\leq -2\beta /\alpha \leq 1,\) then \(f\in B(p,1/\alpha,2^{2\beta /\alpha})\).