Let S denote the class of functions f that are analytic and univalent in the open unit disk U with \(f(0)=f'(0)-1=0\). If g and G are analytic in U, then g is said to be subordinate to G if there exists a Schwarz function w(z) analytic in U with \(g(z)=G(w(z))\) for all z in U. Suppose \(h_ 1\) and \(h_ 2\) are convex univalent functions such that \(h_ 1(U)\) and \(h_ 2(U)\) lie in the right half plane and \(h_ 1(0) = h_ 2(0) =1\). For \(\alpha\geq 0\) the author defines the class \(B(\alpha,h_ 1,h_ 2) = \{f\in S: zf'(z)/(f^{1-\alpha}(z)g^{\alpha}(z))\) is subordinate to \(h_ 2(z)\) in U where \(zg'(z)/g(z)\) is subordinate to \(h_ 1(z)\) in \(U\}\). This class of functions includes several known classes: for example, the class \(B(\alpha,\lambda,\rho)\) introduced by \textit{V. P. Gupta} and \textit{P. K. Jain} [Tamkang J. Math. 7, 117-119 (1976; Zbl 0325.30019)]. The author proves that certain subclasses of \(B(\alpha,h_ 1,h_ 2)\) are invariant under various integral transforms. For example, if \(\alpha >0\), Re c\(>0\) and f belongs to \(B(\alpha,h_ 1,h_ 2)\), then so does \[ [((\alpha +c)/z^ c)\int^{z}_{0}t^{c- 1}f^{\alpha}(t)dt]^{1/\alpha}. \]