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In this article, by employing the hyperbolic tangent function tanhz, a subfamily $\mathcal{S}_{\tanh }^{\ast }$ S tanh ∗ 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 } $ S tanh ∗ 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.

The significant rise in the number of diabetic patients worldwide, as well as the development of new insulin delivery techniques such as inhalation or oral administration which require higher dosages, are expected to increase the demand for recombinant insulin. Current manufacturing technologies will be unable to fulfill the rising demand for inexpensive insulin due to their production capacity limitations and high production costs. Production of therapeutic recombinant insulin requires a suitable host organism with adequate post-translational modification and refolding machinery. E. coli and S. cerevisiae have been used extensively to make recombinant human insulin for medicinal applications. However, transgenic plants are particularly appealing expression systems as they can be used to synthesize huge amounts of insulin for human medicinal purposes. Plant-based expression systems have the potential for high-capacity insulin synthesis at a minimal cost. The significant production of biologically active proinsulin in seeds or leaves with long-term stability provides a low-cost technique to develop proinsulin for both injectable and oral administration. Recently, stem cell therapy is being utilized for the treatment of diabetes, as these cells are capable of differentiating into insulin producing c ells. With the advancement of regenerative medicine research for different chronic diseases, treatment for type 1 diabetes mellitus has been reported. The current review concentrates on several biotechnological attributes applied to the rapid and mass synthesis of biologically active insulin and its analogs in microbes, various types of stem cells and transgenic crops.

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The purpose of this article is to obtain the sharp estimates of the first four initial logarithmic coefficients for the class BTs of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete-Szegö inequality, Zalcman inequality on the logarithmic coefficients and the Hankel determinant H2,1Ff/2 and H2,2Ff/2 for the class BTs with the determinant entry of logarithmic coefficients.