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A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.

Abstract We investigate several sufficient conditions on a function to be convex in one direction or starlike in one direction.

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further, sharp estimate on the generalized Fekete-Szegö functional is also derived. Relevant connections of our results with the existing ones are also made.

The class of functions defined using linear combination of the derivatives of ratio of the normalized analytic function with the identity function is considered in this manuscript. Further, the sharp bounds on the Hankel determinants and estimates on the higher order Schwarzian derivatives for the first three consecutive derivatives are investigated. [ABSTRACT FROM AUTHOR]

The class B¯ B(α) of non-Bazilevič functions was introduced by Obradović. Later, estimates on the second coefficient and Fekete-Szegö functional for normalized analytic functions in the class B¯ B(α) were investigated by Tuneski and Darus. In the present work, sharp estimate on third to eighth coefficients for normalized analytic functions f(z) = z + a2z2 + a3z3 + … ∊ B¯(α) are investigated. Further sharp estimate on the functional ®a2a3 - a4® is also obtained. [ABSTRACT FROM AUTHOR]

We derive sharp upper bound on the initial coefficients and Hankel determinants for normalized analytic functions belonging to a class, introduced by Silverman, defined in terms of ratio of analytic representations of convex and starlike functions. A conjecture related to the coefficients for functions in this class is posed and verified for the first five coefficients. [ABSTRACT FROM AUTHOR]