The goal of this article is to determine sharp inequalities of certain coefficient-related problems for the functions of bounded turning class subordinated with a petal-shaped domain. These problems include the bounds of first three coefficients, the estimate of Fekete-Szegö inequality, and the bounds of second- and third-order Hankel determinants.

The theory of univalent functions has shown strong significance in the field of mathematics. It is such a vast and fully applied topic that its applications exist in nearly every field of applied sciences such as nonlinear integrable system theory, fluid dynamics, modern mathematical physics, the theory of partial differential equations, engineering, and electronics. In our present investigation, two subfamilies of starlike and bounded turning functions associated with a three-leaf-shaped domain were considered. These classes are denoted by ${\mathcal{BT}}_{3l}$ and ${\mathcal{S}}_{3l}^{*}$, respectively. For the class ${\mathcal{BT}}_{3l}$, we study various coefficient type problems such as the first four initial coefficients, the Fekete–Szegö and Zalcman type inequalities and the third-order Hankel determinant. Furthermore, the existing third-order Hankel determinant bounds for the second class will be improved here. All the results that we are going to prove are sharp.

Aim: The study was planned to investigate the phytochemicals, antidiabetic and antioxidant studies of A. consanguineum. Methods: The preliminary studies were performed on crude extract and different solvent fractions. Based on the potency, the chloroform fraction was semi-purified to phyto-fractions CHF-1 – 5. Furthermore, CHF-3 was subjected to isolation of pure compounds using column chromatography. The α-glucosidase, α-amylase and antioxidant assays (DPPH, ABTS, H₂O₂) were performed on all samples. The in-vivo experiments on compounds 1 and 2 were also performed using oral glucose tolerance test. Docking studies were performed on α-glucosidase and α-amylase targets. Results: Among all fractions, the chloroform fraction exhibited excellent activities profile giving IC₅₀ values of 824, 55, 117, 58 and 85 μg/ml against α-glucosidase, α-amylase, DPPH, ABTS and H₂O₂ targets respectively. Among the five semi-purified chloroform phyto-fractions (CHF-1-5), CHF-3 was the leading fraction in activities giving IC₅₀ values of 85.54, 61.19 and 26.58 μg/ml against α-glucosidase, α-amylase and DPPH respectively. Based on the overall potency and physical amount of CHF-3, it was subjected to purification to get compounds 1 and 2. The two compounds were also found potent in in-vitro activities. The observed IC₅₀ values for compound 1 were 7.93, 28.01 and 6.19 μg/ml against α-glucosidase, α-amylase and DPPH respectively. Similarly, the compound 2 exhibited IC₅₀ of 14.63, 24.82 and 7.654 μg/ml against α-glucosidase, α-amylase and DPPH respectively. Compounds 1 and 2 were potent in decreasing the blood glucose levels in experimental animals. Compounds 1 and 2 also showed interactions with the respective enzymes with molecular docking. Conclusions: We can conclude that A. Consanguineum is a rich source of natural antidiabetic agents. Bioguided isolation of compound 1 and 2 showed potential inhibitions in all tested in-vitro antidiabetic targets. Further, both the compounds were also able to decrease the blood glucose levels in experimental animals.

Mirtazapine is a tetracyclic anti-depressant with poor water solubility. The aim of this study was to improve the dissolution rate of mirtazapine by delivering the drug as a liquisolid compact. Central composite design (CCD) was employed for the preparation of mirtazapine liquisolid compacts. In this, the impacts of two independent factors, i.e., excipient ratio (carrier:coating) and different drug concentration on the response of liquisolid system were optimized. Liquisolid compacts were prepared using propylene glycol as a solvent, microcrystalline cellulose as a carrier, and silicon dioxide (Aerosil) as the coating material. The crystallinity of the formulated drug and the interactions between the excipients were examined using X-ray powder diffraction (XRD) and Fourier-transform infrared spectroscopy (FTIR), respectively. The dissolution study for the liquisolid compact was carried out as per FDA guidelines. The results showed loss of crystallinity of the mirtazapine in the formulation and was completely solubilized in non-volatile solvent and equally dispersed throughout the powder system. Moreover, drug dissolution was found to be higher in liquisolid compacts than the direct compressed conventional tablets (of mirtazapine). The liquisolid technique appears to be a promising approach for improving the dissolution of poorly soluble drugs like mirtazapine.

In numerous geometric and physical applications of complex analysis, estimating the sharp bounds of coefficient-related problems of univalent functions is very important due to the fact that these coefficients describe the core inherent properties of conformal maps. The primary goal of this paper was to calculate the sharp estimates of the initial coefficients and some of their combinations (the Hankel determinants, Zalcman’s functional, etc.) for the class of symmetric starlike functions linked with the sigmoid function. Moreover, we also determined the bounds of second-order Hankel determinants containing coefficients of logarithmic and inverse functions of the same class.

In our present study, two subclasses of starlike functions which are symmetric about the origin are considered. These two classes are defined with the use of the sigmoid function and the trigonometric function, respectively. We estimate the first four initial logarithmic coefficients, the Zalcman functional, the Fekete–Szegö functional, and the bounds of second-order Hankel determinants with logarithmic coefficients for the first class ${\mathcal{S}}_{\mathrm{seg}}^{*}$ and improve the obtained estimate of the existing second-order Hankel determinant of logarithmic coefficients for the second class ${\mathcal{S}}_{sin}^{*}$. All the bounds that we obtain in this article are proven to be sharp.