On \(\alpha\)-convex functions of order \(\beta\) of Ruscheweyh type (Q1119770)

Let p be a positive integer, n be any integer greater than -p, \(\beta <1\) and \(\alpha\) be a real number. We prove that if \(f(z)=z^ p+a_{p+1}z^{p+1}+..\). satisfies the condition \[ Re\{(1-\alpha)\frac{(z^ nf(z))^{(n+p)}}{(n+p)(z^{n- 1}f(z))^{(n+p-1)}}+\alpha \frac{(z^{n+1}f(z))^{(n+p+1)}}{(n+p+1)(z^ nf(z))^{(n+p)}})>\beta \quad for\quad | z| <1, \] then \[ Re\frac{(z^ nf(z))^{(n+p)}}{(n+p)(z^{n-1}f(z))^{(n+p-1)}}>\gamma (a,\beta,n,p)\quad for\quad | z| <1, \] where \[ \gamma (\alpha,\beta,n,p)=\{2(n+p+1-\alpha)+2\beta (n+p+1)-3\alpha - \] \[ - [(2(n+p+1-\alpha)+2\beta (n+p+1)-3\alpha)^ 2+16(\alpha -\beta (n+p+1))(n+p+1-\alpha)]^{1}\}/4(n+p+1-\alpha), \] if \(\alpha /(n+p+1)\leq \beta <1/2\) and \(n+p+1>2\alpha\), \[ \gamma (\alpha,\beta,n,p)=\{2\beta (n+p+1)-3\alpha +[(2\beta (n+p+1)-3\alpha)^ 2+8\alpha (n+p+1-\alpha)\quad]^{1/2}\}/4(n+p+1-\alpha), \] if \(1/2\leq \beta <1\) and \(\alpha \neq n+p+1\), and \[ \gamma (\alpha,\beta,n,p)=(1/(3- 2\beta))\quad if\quad 1/2\leq \beta <1\quad and\quad \alpha =n+p+1. \] This result genealizes some results of Goel and Sohi, R. Singh and S. Singh and A. K. Soni.