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    A Comprehensive Analysis of Hermite-Hadamard Type Inequalities via Generalized Preinvex Functions
    (Mdpi, 2021) Tariq, Muhammad; Ahmad, Hijaz; Budak, Hüseyin; Sahoo, Soubhagya Kumar; Sitthiwirattham, Thanin; Reunsumrit, Jiraporn
    The principal objective of this article is to introduce the idea of a new class of n-polynomial convex functions which we call n-polynomial s-type m-preinvex function. We establish a new variant of the well-known Hermite-Hadamard inequality in the mode of the newly introduced concept. To add more insight into the newly introduced concept, we have discussed some algebraic properties and examples as well. Besides, we discuss a few new exceptional cases for the derived results, which make us realize that the results of this paper are the speculations and expansions of some recently known outcomes. The immeasurable concepts and chasmic tools of this paper may invigorate and revitalize additional research in this mesmerizing and absorbing field.
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    Integral inequalities for n-polynomial s-type preinvex functions with applications
    (Wiley, 2021) Ihsan Butt, Saad; Budak, Huseyin; Tariq, Muhammad; Nadeem, Muhammad
    In this present paper, we introduce the idea and concept of n-polynomial s-type preinvex functions. We elaborate and investigate the algebraic properties of the newly introduced definition and discuss their connections and relations with convex functions. We find the new sort of Hermite-Hadamard inequality via a newly introduced definition. Furthermore, some refinements of Hermite-Hadamard inequality are given. Finally, we investigated some applications via this newly introduced definition. The results obtained in this paper can be viewed as significant improvement of previously known results.
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    Radiation efficiency and nitrogen fertilizer impacts on sunflower crop in contrasting environments of Punjab, Pakistan
    (Springer Heidelberg, 2018) Nasim, Wajid; Ahmad, Ashfaq; Amin, Asad; Tariq, Muhammad; Awais, Muhammad; Saqib, Muhammad; Ali, Shahzad
    Sunflower (Helianthus annuus L.) is the leading non-conventional oilseed crop in Pakistan. Nitrogen fertilizer can affect plant growth and productivity by changing canopy size which has an effect on the radiation use efficiency (RUE) of the crop. The response of sunflower hybrids in terms of phenology, fraction of intercepted radiation (F-i), and RUE to nitrogenous rates (0, 60, 120, 180, and 240 kg ha(-1)) was studied in three field experiments conducted in three various environments: Multan (arid), Faisalabad (semi-arid), and Gujranwala (sub-humid) during spring seasons 2008 and 2009. The treatments were laid out according to a randomized complete block design with split plot arrangements, keeping the sunflower hybrids in main plots and nitrogen rates in subplots, and replicated three times. The results showed Hysun-38 took a maximum number of days to anthesis (101) as compared to Pioneer-64A93 (100) and Hysun-33 (99). The mean values of F-i were 0.850, 0.903, and 0.978, and the estimated values of RUE for total aboveground dry matter were 2.14, 2.47, and 2.65 g MJ(-1) at experimental locations of Multan, Faisalabad, and Gujranwala, respectively. The values of RUE for grain yield (RUEGY) were 0.78, 0.98, and 1.26 g MJ(-1) at experimental locations of Multan, Faisalabad, and Gujranwala, respectively. The average RUEGY values over three locations were 2.61, 2.60, 2.43, and 2.36 g MJ(-2) in N-4 (180 kg ha(-1)), N-5 (240 kg ha(-1)), N-3 (120 kg ha(-1)), and N-2 (60 kg ha-1) treatments, respectively. Increasing rates of N increased RUEGY over the standard treatment N-3 (120 kg N ha(-1)); however, the averaged values over three locations were 1.22, 1.08, 0.99, and 0.92 g MJ(-2) in N-4, N-5, N-3, and N-2 treatments, respectively. Therefore, optimum water and N doses are important for attaining higher RUE, which may enhance sunflower growth and yield.