In this article, by employing the hyperbolic tangent function tanhz, a subfamily $\mathcal{S}_{\tanh }^{\ast }$Stanh∗ of starlike functions in the open unit disk $\mathbb{D}\subset \mathbb{C}$D⊂C : $$\begin{aligned} \mathbb{D}= \bigl\{ z:z\in \mathbb{C} \text{ and } \vert z \vert < 1 \bigr\} \end{aligned}$$D={z:z∈C and |z|<1} is introduced and investigated. The main contribution of this article includes derivations of sharp inequalities involving the Taylor–Maclaurin coefficients for functions belonging to the class $\mathcal{S}_{\tanh }^{\ast } $Stanh∗ of starlike functions in $\mathbb{D}$D . In particular, the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functionals, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.

In the present paper, by using the concept of convolution and q-calculus, we define a certain q-derivative (or q-difference) operator for analytic and multivalent (or p-valent) functions. This presumably new q-derivative operator is an extension of the known q-analogue of the Ruscheweyh derivative operator. We also give some interesting applications of this q-derivative operator for multivalent functions by using the method of differential subordination. Relevant connections with a number of earlier works on this subject are also pointed out.

Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in this paper, we use two q-operators $\mathbb{T}(a,b,c,d,e,y{D}_x)$ and $\mathbb{E}(a,b,c,d,e,y{\theta }_x)$ to derive two potentially useful generalizations of the q-binomial theorem, a set of two extensions of the q-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the q-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

In this article, we introduce a new subclass of analytic functions utilizing the idea of Mittag-Leffler type Poisson distribution associated with the Janowski functions. Further, we discuss some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szeg inequality, and partial sums for this newly defined class.

In this review paper, our aim is to study the current research progress of q-difference equations for generalized Al-Salam–Carlitz polynomials related to theta functions and to give an extension of q-difference equations for q-exponential operators and q-difference equations for Rogers–Szegö polynomials. Then, we continue to generalize certain generating functions for Al-Salam–Carlitz polynomials via q-difference equations. We provide a proof of Rogers formula for general Al-Salam–Carlitz polynomials and obtain transformational identities using q-difference equations. In addition, we gain $U(n+1)$-type generating functions and Ramanujan’s integrals involving general Al-Salam–Carlitz polynomials via q-difference equations. Finally, we derive two extensions of the Andrews–Askey integral via q-difference equations.