Let \(S:*(\alpha),\;K(\alpha)\) be the class of starlike and convex functions of order \(\alpha\), analytic in the open unit disk. Define the convolution of \(f_ 1 * f_ 2 (z)=\sum:\infty_{n=0} a_{1,n+1}a_{2,n+1}z:{n+1},\;f_ i (z)= \sum:{\infty}_{n=0}a_{i,n+1}z:{n+1},\;i=1,\;2.\) Let \((\lambda)_ n\) be the Pochhammer symbol defined by \((\lambda)_ 0=1,\;(\lambda)_ n=\lambda (\lambda +1)\cdots(\lambda +n-1),\;n=1,\;2,\cdots\) and \(\phi (a,c;z)=\sum:\infty_{n=0}{(a)_ n\over(c)_ n}z:{n+1},\;c\neq0,\;-1,\;- 2,\;\cdots\). \(\phi (a,c;z)\) is the incomplete beta function \(z_ 2F_ 1(1,a;c;z)\). Define the linear operator \(L(a,\;c)f(z)=\phi (a,c;z)*f(z)\) [\textit{B. C. Carlson} and \textit{D. B. Shaffer}, SIAM J. Math. Anal. 15, 737-745 (1984; Zbl 0567.30009)] and the integral operator \[ I:{\alpha,\;\beta,\;\eta}_{0,\;z} f(z)={z:{-\alpha - \beta}\over{\Gamma (\alpha)}}\int:z_ 0 {(z-\zeta):{\alpha-1}}_ 2F_ 1(\alpha+\beta,\;- \eta\;;\alpha;1- {\zeta\over z})f(\zeta)d(\zeta). \] The authors define the fractional operator \[ J:{\alpha,\;\beta,\;\eta}_{0,\;z} f(z)={{\Gamma (2-\beta)\Gamma (2+\alpha+\eta)}\over{\Gamma (2-\beta+\eta)}} z:\beta I:{\alpha,\;\beta,\;\eta}_{0,\;z} f(z) \] and note \[ J:{\alpha,\;\beta,\;\eta}_{0,\;z} f(z)=L(2,2-\beta) L(2- \beta+\eta, 2+\alpha+\eta) f(z). \] They then prove a number of inclusion relations for this operator between the classes \(S:*(\alpha),\;K(\alpha)\) as well as the prestarlike functions of order \(\alpha\;(\alpha\leq 1),\;R(\alpha)\). Examples are: (1) If \(\alpha>0,\;0\leq\beta0,\;0\leq\beta<2,\) and \(\eta\) is real, then \[ L(2+\alpha+\eta,2-\beta+\eta)J:{\alpha,\;\beta,\;\eta}_{0,\;z} K(\beta /2)\subset R(\beta/ 2). \]