Polar Derivatives Research Articles

Spinacia oleracea L. Baby Leaves as a Source of Bioactive Principles: The Chemical Profiling of Eco-Sustainable Extracts by Using LC-ESI/HRMS- and 1H NMR-Based Metabolomics.

Spinacia oleracea L. cultivar platypus leaves are identified as a functional food due to their nutrient composition which promotes health beyond basic nutrition. Considering the increasing use of food supplements, S. oleracea baby leaves have been extracted by maceration, solid-liquid dynamic extraction (SLDE)-Naviglio, and ultrasound-assisted extraction (UAE) using EtOH and EtOH:H2O mixtures. The analysis of the extracts by using LC-ESI/HRMSMS revealed 42 compounds (flavonoids, polar lipid derivatives, and 20-hydroxyecdysone), along with primary metabolites, detected by NMR analysis. A principal component analysis (PCA) of LC-ESI/HRMS and NMR data was performed, revealing how 20-hydroxyecdysone and flavonoids, the specialized metabolites mainly responsible for the biological activity of S. oleracea leaves, occurred in the highest amount in the EtOH and EtOH:H2O (70:30, v/v) extracts obtained by SLDE-Naviglio extraction. 20-hydroxyecdysone was also quantified in all the extracts via LC-ESI/QTrap/MS/MS using the Multiple Reaction Monitoring (MRM) method. The EtOH extracts obtained by SLDE-Naviglio and maceration showed the highest content (82.16 and 81.27 mg/g extract, respectively). The total phenolic content (118.35-206.60 mg GAE/g), the flavonoid content (10.90-41.05 mg rutin/g), and the Trolox Equivalent Antioxidant Capacity (TEAC) (1.63-2.05 mM) of the extracts were determined. The EtOH:H2O (70:30, v/v) extract analyzed by using SLDE-Naviglio showed the highest phenolic and flavonoid content and radical scavenging activity.

Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros

Abstract In this article, we extend inequalities concerning the polar derivative of a polynomial to integral mean for the class of polynomials with s-fold zero at the origin and the remaining zeros inside some closed disk of radius k k for k ≥ 1 k\ge 1 and k ≤ 1 k\le 1 , and we thereby obtain bounds that depend on some coefficients of the underlying polynomial. Our results not only extend some known polynomial inequalities but also reduce some interesting results in particular cases. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to previously established results.

Polar Thiazole Derivative-Induced Second Harmonic Response in Hybrid Bismuth Chlorate with Reversible Photochromism.

Inorganic-organic hybrid bismuth halides demonstrate great prospect in the field of second-order nonlinear optical (NLO) crystals because the ns2 electron on Bi(III) could lead to large molecular polarization and high second harmonic generation (SHG) coefficient on a noncentrosymmetric structure. However, researchers cannot yet control the effective arrangements of the bismuth halide functional motif, which results in SHG-active hybrid bismuth halides being rare. Herein, thiazole derivatives with a polar donor-π-acceptor system are designed to explore hybrid bismuth halide NLO crystals. The protonated 2-(4-hydroxyphenyl) thiazole (denoted as hpt) interacts with the (BiCl6) motif via H···Cl hydrogen bonds to prevent antiparallel arrangements, which therefore enhance the NLO property. (Hhpt)3[BiCl6] crystallizes in the monoclinic polar P21 space group, which exhibits a strong SHG response and reversible photochromic behavior. Structural analysis and theory calculations reveal that the synergistic effect between the polar thiazole derivative and the distorted inorganic [BiCl6]3- motif is responsible for the SHG response. This is the first report of polar thiazole derivative-induced SHG-active hybrid bismuth halide coupled with reversible photochromism.

Lq-norm inequalities for the generalized derivative of a polynomial

In this paper, we aim to present some Turan-type inequalities in Lq-norm for the generalized derivative of a polynomial. We further extend these results to the generalized polar derivative. Our results besides generalizing various known results also include other inequalities as special cases.

On Bernstein and Turán-type integral mean estimates for polar derivative of a polynomial

Let p(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p(z)$\\end{document} be a polynomial of degree n having no zero in |z|<k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|z|< k$\\end{document}, k≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k\\leq 1$\\end{document}, then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max|z|=1|p′(z)|≤n1+knmax|z|=1|p(z)|,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\max _{|z|=1}|p'(z)|\\leq \\frac{n}{1+k^{n}}\\max _{|z|=1}|p(z)|, $$\\end{document} provided |p′(z)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|p'(z)|$\\end{document} and |q′(z)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|q'(z)|$\\end{document} attain their maxima at the same point on the circle |z|=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|z|=1$\\end{document}, where q(z)=znp(1z‾)‾.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q(z)=z^{n}\\overline{p\\bigg(\\frac{1}{\\overline{z}}\\bigg)}. $$\\end{document} In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results.

On the Erdös–Lax-Type Inequalities for Polynomials

Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject.

Analysis of scopolamine and its related substances by means of high-performance liquid chromatography

AbstractThe retention behaviour of scopolamine (hyoscine) and its related compounds (norhyoscine, atropine, homatropine, and noratropine) was investigated on the silica-based HPLC stationary phase. The retention of investigated tropane alkaloids was interpreted by using the Soczewiński-Wachtmeister equation. A high correlation between the retention parameter (log k) and lipophilicity (log P) (R = 0.9923) confirms the significant influence of hydrophobic interactions on the retention behaviour of the aforementioned compounds. It was found that by increasing the acetonitrile fraction, a decrease in retention of the more polar epoxide derivatives (scopolamine, norhyoscine) and an increase in retention of the more lipophilic derivatives (atropine, noratropine, homatropine) is obtained. The best separation of the tropane alkaloids was achieved by a simple procedure that involved a mobile phase composed of acetonitrile and 40 mM ammonium acetate/0.05% TEA, pH 6.5; 50:50 v/v. Selected conditions were assumed for the determination of scopolamine hydrochloride in the eye drops (Scopolamini hydrobromidum 0.25%). The method was validated and it was found as selective, sensitive, precise, accurate, and robust for the further qualitative analysis of the scopolamine-related compounds.

On some inequalities concerning polynomials with restricted zeros

Abstract In this work we prove some new Bernstein-type inequalities for the polynomials with restricted zeros. Our results strengthen some recently proved Erdős–Lax and Turán-type inequalities by Kumar and others. We further extend the obtained results to the polar derivative of a polynomial.

Some Lp norm involving the polar derivative of a polynomial

Let [Formula: see text], [Formula: see text], is a polynomial of degree [Formula: see text] having all zeros in [Formula: see text], [Formula: see text], then for [Formula: see text] with [Formula: see text], [Formula: see text] and for each [Formula: see text] with [Formula: see text]; it was proved by (N. A. Rather et al., Some [Formula: see text] inequalities involving the polar derivative of a polynomial, Appl. Math. E-Notes 14 (2014) 116–122) [Formula: see text] [Formula: see text] In this paper, we establish some refinements and generalizations of above and some known polynomial inequalities concerning the polar derivative of a polynomial.

Targeted Offline Two-Dimensional HPLC and UHPLC-Orbitrap-MS Combined with Molecular Networking Reveal the Effect of Processing on Chemical Constituents of Xuetong (the Stem of Kadsura heteroclita)

Xuetong, the dried stem of Kadsura heteroclita (Roxb.) Craib, is a traditional Tujia medicine extensively used to treat rheumatoid arthritis (RA). All traditional Chinese medicines (TCMs) necessitate a processing stage called “Paozhi” before clinical application. However, there is a dearth of research concerning the processing methods employed for Xuetong. To investigate the impact of vinegar and wine processing on the chemical constituents of Xuetong, this study devised a targeted offline two-dimensional (2D) high-performance liquid chromatography (HPLC) and ultra-high-performance liquid chromatography-orbitrap mass spectrometry (UHPLC-orbitrap-MS) method. By incorporating various MS data-processing techniques, such as molecular networking technology, fragment-ion similarity searching (FISh), online and offline database matching, and fragmentation pattern analysis, a total of 158 components were identified in Xuetong. Among them, 14 were verified by comparison with the reference standards. Notably, aside from triterpenoids and lignans, catechin derivatives were found to be the predominant constituents of Xuetong, and their levels exhibited a significant decrease following processing. This method significantly improved peak capacity and resolution, overcoming the limitations of 1D LC in simultaneously analyzing highly polar catechin derivatives and less polar triterpenoids and lignans. Moreover, the developed method shows promise for Xuetong’s quality control.

NEW REGION FOR ZEROS OF A POLAR DERIVATIVE OF POLYNOMIAL

Let P(z) := Pn j=0 aj z j be a polynomial of degree n. Then by Cauchy’s Classical result, all zeros of P(z) lie in |z| ≤ 1 + max 0≤j≤n−1 | aj an | . In this paper, we shall extend such results to polar derivative of an algebraic polynomial.

Generalizations of some Bernstein-type Inequalities for the Polar Derivative of a Polynomial

In this paper, we establish some new Bernstein-type bounds for the polar derivative of constrained polynomials on the unit circle in the plane. The obtained results sharpen some known estimates for the ordinary derivative of polynomials as special cases

GENERALIZED Lᵖ INEQUALITIES FOR THE POLAR DERIVATIVE OF POLYNOMIAL

Let P(z) be a polynomial of degree n and let α be any real or complex number, then the emanant or polar derivative of p(z) is denoted by Dαp(z) and defined as Dαp(z) = np(z) + (α − z)p 0 (z). The polynomial Dαp(z) is of degree at most n − 1 and it generalises the ordinary derivative p 0 (z) of p(z) in the sense that limα→∞ Dαp(z) α = p 0 (z). In this paper, we prove some L p inequalities for the emanant of the polynomial having all its zeros in prescribed disk. Our results generalize the earlier known results.

Efficient copolymerization of ethylene with norbornene mediated by phosphine-sulfonate palladium catalysts

A series of tBu-/Ar-substituted phosphine-sulfonate palladium catalysts Pd1-Pd6 {[κ2(P, O)-(tBu)ArP(C6H4SO2O)]Pd(Me)(DMSO)} (1: Ar = Ph; 2: Ar = 4-OMe-C6H4; 3: Ar = 4-CF3-C6H4; 4: Ar = 2-OMe-C6H4; 5: Ar = 2,5- (OMe)2-C6H3; 6: Ar = 2,4-(OMe)2-C6H3) were prepared to copolymerize ethylene with norbornene and its polar derivatives. Installing electron-donating OMe substituent on the nonchelating P-bound aryl was proven to enhance the properties of catalysts. The position and number of OMe substituent significantly influenced the copolymerization results. The E-NB copolymers generated from these catalysts showed both high comonomer incorporations (≥19.3 mol%) and high copolymer molecular weights (≥24.6 kg⋅mol−1), which has been proven to be difficult to achieve using this catalyst system. And for the obtained copolymers, their norbornene incorporations and Tg values conformed to a linear relationship. More importantly, these catalysts can efficiently mediate the copolymerizations of ethylene and polar norbornenes, along with high activities (up to 1.0 × 106 g⋅mol−1⋅h−1), high molecular weights (up to 182 kg⋅mol−1), and high incorporations (up to 36.7 mol%). This work provided an efficient method to synthesis cyclic olefin copolymer using late transition metal catalysts.

Integral Inequalities for the Polar derivative and the generalized Polar derivative of complex Polynomials

For a polynomial P(z) of degree n , having all zeros in |z|≤1, Malik [11] proved that for each q&gt;0, \(n [∫_0^{2π} |P(e^{iθ})|^q dθ] ^{1⁄q} ≤[∫_0^{2π} | 1 + e^{iθ} |^q dθ]^{1⁄q} \underset{|z|=1}{max} |P^ʹ (z)|\). In this paper we generalize the above inequality to polar derivative and generalized polar derivative, which as special cases include several known results in this area.

Generalized Turan-type Inequalities for Polar Derivative of a Polynomial

Let $P(z)=a_0+\sum\limits_{\nu=\mu}^na_{\nu}z^{\nu}$, $1\leq\mu\leq n$, be a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $k\geq 1$. We obtain an improvement and a generalization of an inequality in polar derivative proved by Somsuwan and Nakprasit [1]. Further, we also extend a result proved by Chanam and Dewan [2] to its polar version. Besides, our results are also found to generalize and improve some known inequalities.

On a Generalization of an Operator Preserving Turán-Type Inequality for Complex Polynomials

Let W(zeta)=(a_{0}+a_{1}zeta+...+a_{n}zeta^{n}) be a polynomial of degree n having all its zeros in mathbb{T}_{k}cupmathbb{E}^{-}_{k}, kgeq 1, then for every real or complex number alpha with |alpha|geq 1+k+k^{n}, Govil and McTume [7] showed that the following inequality holds\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\max\\limits_{\\zeta\\in\\mathbb{T}_{1}}|D_{\\alpha}W(\\zeta)|\\geq n\\left(\\frac{|\\alpha|-k}{1+k^{n}}\\right)||W||+n\\left(\\frac{|\\alpha|-(1+k+k^{n})}{1+k^{n}}\\right)\\min\\limits_{\\zeta\\in\\mathbb{T}_{k}}|W(\\zeta)|.$$\\end{document}In this paper, we have obtained a generalization of this inequality involving sequence of operators known as polar derivatives. In addition, the problem for the limiting case is also considered.

Boundedness in the Bloch Space of Symmetric Domain for a Class of Multi-Valent Meromorphic Functions Given by a Fractional Integral

Convolution operators have profited in various areas of science. They are utilized in the investigations of computing techniques. A new convolution operator linked to a specific class of multi-valent meromorphic functions in the punctured unit disk (symmetric domain) is formulated. This analysis uncovers certain properties on the connections as well as the power series. We study a novel class of holomorphic functions concerning the recommended new operator. The second part of the outcome concerns the boundedness of the suggested difference structure given by the proposed operator. We focus on the Bloch space of meromorphic functions in the open unit disk. In this case, we use the spherical derivative. To obtain the maximum value of the polar derivative of the polynomials created by their partial sums, we use their partial sums as applications of the suggested operator.

Lγ Inequalities for the Polar Derivative of Polynomials

In this paper, firstly, we obtain an inequality in Lγ analogue concerning the polar derivative for a polynomial p(ξ) = Xm ν=0 cνξν of degree m having no zero in |ξ| &lt; r, r ≥ 1 proved by Govil et al. [15]. Secondly, we also prove Lγ version for the polar derivative of an ordinary inequality for a polynomial having all its zeros in |ξ| ≤ r, r ≤ 1 proved in that same paper. Our results generalize and improve some known inequalities.

On some operator preserving inequalities between polynomials

In this paper, some generalizations of lower bound estimates for the maximum modulus of the [Formula: see text]th polar derivative [Formula: see text], where [Formula: see text], [Formula: see text] of polynomials are established under the assumption that one of the underlying polynomials [Formula: see text] and [Formula: see text] does not vanish in [Formula: see text]. The obtained estimates include several known lower bounds for polar derivatives as special cases.