We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0<\alpha<1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

Candida albicans is a major fungal pathogen of humans, accounting for 15% of nosocomial infections with an estimated attributable mortality of 47%. C. albicans is usually a benign member of the human microbiome in healthy people. Under constant exposure to highly dynamic environmental cues in diverse host niches, C. albicans has successfully evolved to adapt to both commensal and pathogenic lifestyles. The ability of C. albicans to undergo a reversible morphological transition from yeast to filamentous forms is a well-established virulent trait. Over the past few decades, a significant amount of research has been carried out to understand the underlying regulatory mechanisms, signaling pathways, and transcription factors that govern the C. albicans yeast-to-hyphal transition. This review will summarize our current understanding of well-elucidated signal transduction pathways that activate C. albicans hyphal morphogenesis in response to various environmental cues and the cell cycle machinery involved in the subsequent regulation and maintenance of hyphal morphogenesis.

The promotion of plant growth and suppression of plant disease using beneficial microorganisms is considered an alternative to the application of chemical fertilizers or pesticides in the field. In this study, a coconut-scented antagonistic Trichoderma strain LZ42, previously isolated from Genoderma lucidum-cultivated soil, was investigated for biostimulatory and biocontrol functions in tomato seedlings. Morphological and phylogenetic analyses suggested that strain LZ42 is closely related to T. atroviride. Tomato plants showed increased aerial and root dry weights in greenhouse trials after treatment with T. atroviride LZ42 formulated in talc, indicating the biostimulatory function of this fungus. T. atroviride LZ42 effectively suppressed Fusarium wilt disease in tomato seedlings, with an 82.69% control efficiency, which is similar to that of fungicide treatment. The volatile organic compounds (VOCs) emitted by T. atroviride LZ42 were found to affect the primary root growth direction and promote the root growth of tomato seedlings in root Y-tube olfactometer assays. The fungal VOCs from T. atroviride LZ42 were observed to significantly inhibit F. oxysporum in a sandwiched Petri dish assay. SPME-GC-MS analysis revealed several VOCs emitted by T. atroviride LZ42; the dominant compound was tentatively identified as 6-pentyl-2H-pyran-2-one (6-PP). 6-PP exhibited a stronger ability to influence the direction of the primary roots of tomato seedlings but not the length of the primary roots. The inhibitory effect of 6-PP on F. oxysporum was the highest among the tested pure VOCs, showing a 50% effective concentration (EC50) of 5.76 μL mL-1 headspace. In conclusion, T. atroviride LZ42, which emits VOCs with multiple functions, is a promising agent for the biostimulation of vegetable plants and integrated management of Fusarium wilt disease.

Image stitching is a crucial aspect of image processing. However, factors like perspective and environment often lead to irregular shapes in stitched images. Cropping or completion methods typically result in substantial loss of information. This paper proposes a method for rectifying irregularly images into rectangles using deformable meshes and residual networks. The method utilizes a convolutional neural network to quantify rigid structures of images. Choosing the most suitable mesh structure based on the extraction results, offering options such as triangular, rectangular, and hexagonal. Subsequently, the irregularly image, predefined mesh structure, and predicted mesh structure are input into a wide residual neural network for regression. The loss function comprises local and global, aimed at minimizing the loss of image information within the mesh and global target. This method not only significantly reduces information loss during rectification but also adapting to different images with various rigid structures. Validation on the DIR-D dataset shows this method outperforms state-of-the-art methods in image rectification.

Supramolecular solvent (SUPRAS) has received increasing interest as an innovative, efficient, and green solvent for the effective extraction and separation of bioactive compounds from natural resources. However, studies on the use of SUPRAS for the extraction of phenolic components from plants, which are highly valued in food products due to their excellent antioxidant capacity, are still lacking. The present study developed a green ultrasound-assisted and efficient SU-PRAS method to simultaneously determine three phenolic acids in Prunella vulgaris by high-performance liquid chromatography (HPLC) and the experimental parameters were optimized in detail. The efficiency and antioxidant properties of the phenolic compounds obtained by different extraction methods were also compared. Under the optimal conditions, the extraction efficiency of SUPRAS, prepared with octanoic acid reverse micelles dispersed in ethanol-water, was substantially higher than conventional organic reagents. Moreover, this SUPRAS method also showed more significant antioxidant capacities. The confocal laser scanning microscopy (CLSM) images presented the spherical droplet structure of SUPRAS (A well-defined circle), which coincided with the position of the fluorescence position of phenolic acids. The phenolic acids were encapsulated in the vesicles by the SUPRAS droplets, which indicated the efficient extraction capacities of SUPRAS for the target compounds. Finally, combined with the CLSM method, the mechanism of the proposed method was further clarified, and the superior extraction performance of SUPRAS was theoretically demonstrated through molecular dynamics simulations. In comparison with conventional methods, the higher extraction efficiency of SUPRAS can be illustrated by the larger solvent contact surface area, more types of hydrogen bonds between extractants and supramolecular solvents, and stronger and more stable interaction forces. The results of the theoretical studies are in rational agreement with the experimental outcome.