Some subclasses of uniformly convex functions involving certain fractional calculus operator (Q2910469)

In this paper, the authors define a class \(MT_{\mu ,\gamma ,\eta }(\alpha ,\beta ,\lambda )\) of univalent functions NEWLINE\[NEWLINE\displaystyle f(z)=z-\sum_{k=2}^{\infty }a_{k}z^{k}, \quad a_{k}\geq 0,\left| z\right| <1NEWLINE\]NEWLINE with negative coefficients such that the linear transformation \( G_{0,z}^{\mu ,\gamma ,\eta }f(z)\) defined by Equation (1.8) in the article satisfies the Inequality (1.10). As they remark on the top of the page 238, among other things, they obtain some characterization properties, coefficient estimates and distortion inequalities. One should note that the linear operator NEWLINE\[NEWLINEF(z)=Lf(z)=(1- \lambda )G_{0,z}^{\mu ,\gamma ,\eta }f(z)+\lambda zG_{0,z}^{\mu ,\gamma ,\eta }f(z)NEWLINE\]NEWLINE is one-to-one and maps the class \(MT_{\mu ,\gamma ,\eta }(\alpha ,\beta ,\lambda )\) onto the class \(S_{p}(\alpha ,\beta )\cap T\) of univalent functions with negative coefficients defined by the inequality (1.4). A careful calculation shows that the inequality (1.4) is equivalent to the inequality NEWLINE\[NEWLINE\mathrm{Re}\left\{z\frac{^{f^{^{\prime }}(z)}}{f(z)}\right\}>\frac{ \alpha +\beta }{1+\beta }.NEWLINE\]NEWLINE Therefore, the class \(S_{p}(\alpha ,\beta )\cap T \) is equal to the class \(T^{\ast }(\frac{\alpha +\beta }{1+\beta })\) of starlike functions of order \(\frac{\alpha +\beta }{1+\beta }\) with negative coefficients introduced by \textit{H. Silverman} [Proc. Am. Math. Soc. 51, 109--116 (1975; Zbl 0311.30007)]. It is well known that a linear transformation maps convex sets to convex sets and carries the extreme points to extreme points. Hence, almost all the results and the referred results in this paper follow from the known ones. For example, Theorem 2.1 and Theorem 5.1 in this paper follow from Theorem 2 and Theorem 9 of H. Silverman, respectively.NEWLINENEWLINEMany authors do not analyze defining properties of the classes of functions with negative coefficients. As a result, they think that they obtain some new results but, very often, these results are equivalent to previous ones. Because of this, I urge all the authors who work and have worked on classes of univalent functions with negative coefficients to reconsider their results whether or not there are some duplications in their work.