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Generalization of Mixed Means and Related Results

Thesis Info

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Author

Khan, Khuram Ali

Supervisor

Josip Pecaric

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2008

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2051/1/1718S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726257502

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Inequalities lie at the heart of a great deal of mathematics. G. H. Hardy reported Harald Bohr as saying ‘all analysts spend half their time hunting through the literature for inequalities which they want to use but cannot prove’. Inequalities involving means open many doors for analysts e.g generalization of mixed means fallouts the refinements to the important inequalities of Holder and Minkowski. The well known Jensen’s inequality asserts a remarkable relation among the mean and the mean of function values and any improvement or refinements of Jensen’s inequality is a source to enrichment of monotone property of mixed means. our aim is to utilize all known refinements of Jensen’s inequality to give the re- finements of inequality among the power means by newly defined mixed symmetric means. In this context, our results not only ensures the generalization of classical but also speak about the most recent notions (e.g n-exponential convexity) of this era. In first chapter we start with few basic notions about means and convex functions. Then the classical Jensen’s inequality and the historical results about refinements of Jensen’s inequality are given from the literature together with their applications to the mixed symmetric means. In second chapter we consider recent refinements of Jensen’s inequality to refine inequality between power means by mixed symmetric means with positive weights under more comprehensive settings of index set. A new refinement of the classical Jensen’s inequality is also established. The Popovicui type inequality is generalized using green function. Using these refinements we define various versions of linear functionals that are positive on convex functions. This step ultimately leads us to viiviii the important and recently revitalized area of exponential convexity. Mean value theorems are proved for these functionals. Some non-trivial examples of exponential convexity and some classes of Cauchy means are given. These examples are further used to show monotonicity in defining parameters of constructed Cauchy means. In third chapter we develop the refinements of discrete Jensen’s inequality for con- vex functions of several variables which causes the generalizations of Beck’s results. The consequences of Beck’s results are given in more general settings. We also gen- eralize the inequalities of H ̈older and Minkowski by using the Quasiarithmetic mean function. In forth chapter we investigate the class of self-adjoint operators defined on a Hilbert space, whose spectra are contained in an interval. We extend several re- finements of the discrete Jensen’s inequality for convex functions to operator convex functions. The mixed symmetric operator means are defined for a subclass of positive self-adjoint operators to give the refinements of inequality between power means of strictly positive operators. In last chapter, some new refinements are given for Jensen’s type inequalities in- volving the determinants of positive definite matrices. Bellman-Bergstrom-Fan func- tionals are considered. These functionals are not only concave, but superlinear which is a stronger condition. The results take advantage of this property.
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پنڈت کشن پرشاد کول

پنڈت کشن پرشاد کول
پنڈت کشن پرشادکول لکھنؤ کے رہنے والے اور لکھنؤ کی پُرانی تہذیب اور روایات کی بڑی حسین یادگار تھے، اُردو زبان کے نکتہ سنج ونکتہ شناس، بلند پایہ ادیب اور بڑے صحیح الخیال اورنیک دل بزرگ تھے۔اُردو زبان اوراس کے ساتھ وابستہ کلچر اورتہذیب کے ساتھ اُن کوعشق تھا۔ یوپی میں اُردو کو علاقائی زبان بنانے کے لیے دستخطی مہم کی کامیابی زیادہ ترانھیں کی کوششوں کانتیجہ تھی۔اُردو سے متعلق ہرتحریک میں وہ سرگرمی سے حصّہ لیتے تھے۔ چنانچہ وفات سے کچھ دن پہلے ہی انھوں نے بہار میں اردو کانفرنس کی صدارت کی تھی اوراس میں بڑا اچھا خطبہ پڑھا تھا۔ اگرچہ وہ اب خود دنیا میں نہیں ہیں لیکن اپنے کردار وعمل اوراپنے اخلاق کاایک ایسا نقش چھوڑگئے ہیں کہ نئی نسلیں اُن سے بہت کچھ سبق حاصل کرسکتی ہیں۔ [ فروری ۱۹۵۵ء]

 

اسلام كے فوجداری نظام كا ضابطہ قسامت

e Qasama Doctrine of Islamic Criminal Law The mashroom-growth like blind murder cases, have, now a days confused and perplexed the law-enforcing agencies the reason is that such murder-cases are taken in hand and tried to be dealt with the common criminal procedures The criminal in such a case leaving no clue thereto succeed in detracting the police. As a result the FIR is lodged against an anonymous 'accused' afterwards and the case is filed because of the non-availability of required proof. Contrary to the above Islam introduces the procedure cf Qasama _ which literally means administring an oath which in juristic terminology applied to a way and process where some persons are held responsible in a blind murder for an oath in words, that; By Allah! Neither they have committed the murder nor they noticed the culprit. In case of refusal they are adjudicated for Qisasand for the payent of Diyat in vice versa. Historically Qasama procedure is traced back to pre-lslamic tribal-law which were then, afterwards modified and re-enforced by the Prophet (SAW) and his Khulafa. With the exception of some minor juristic controversies regarding the structure and framework of Qasama procedure multitude of muslim jurists hold it a valid way for the adjudication of a blind-murder. It is with all regrets that-lslamic Ideological council ( HC) despite its introduction .

Design of Protein-Targeted Organometallic Complexes As Anticancer Agents.

DNA is considered as the ultimate target of platinum based anticancer drugs which are widely used in clinics but the toxicity and resistance induced by these compounds have halted their success. In recent past, proteins or enzymes have been explored as alternate targets for metal-based anticancer agents. These enzymes or proteins are involved in metabolic pathways associated with cancer development. These include transferrin, albumin, kinase, cathepsin B, thioredoxin reductase, plectin, carbonic anhydrase and histone deacetylase etc. Many compound classes of metal complexes have been investigated against such targets. The ruthenium and osmium complexes of pyridine-2-carbothioamides (PCAs) stabilized by η6-arene ring were introduced as orally administrable anticancer agents with potential to bind with the histone proteins to interrupt the chromatin activity (Chemical Science., 2013, 4, 1837–1846). Recently, in vivo examination of these compounds revealed selective binding to plectin and they termed as plecstatin (Angewandte Chemie International Edition., 2017, 56, 8267-8271). In this doctoral thesis, PCA ligands were functionalized with groups which can bind to specific enzymes or proteins such as carbonic anhydrase and histone deacetylase. The new PCA ligands were then converted to their respective organometallic compounds of Ru(II), Os(II), Rh(III) and Ir(III). All novel PCAs and their corresponding complexes were evaluated for their cytotoxic potential against different cancer cell. The organometallic compounds were studied for their hydrolytic stability as well as their interactions with biomolecules such as amino acids and proteins by using a range of biophysical methods. For structure activity relationships study, a series of N-phenyl substituted pyridine-2- carbothiamides (PCAs) were obtained by systematically varying the substituents at the phenyl ring. The PCAs were then converted to their corresponding RuII(η6-pcymene) complexes. In preliminary examination, these metal based compounds were studied for their acidic and hydrolytic stability. In cytotoxic assay, the lipophilic PCAs 1–4 showed cytotoxicity in the low micromolar range and 6 was the most potent compound of the series with an IC50 value of 1.1 μM against HCT116 colon cancer cells. These observations were correlated with calculated octanol/water partition coefficient (clogP) data and quantitative estimated druglikeness. A similar 17 trend as for the PCAs was found in their Ru complexes, where the complexes with more lipophilic ligands proved to be more cytotoxic in all tested cell lines. In general, the PCAs and their organoruthenium derivatives demonstrated excellent drug-likeness and cytotoxicity with IC50 values in the low micromolar range, making them interesting candidates for further development as orally active anticancer agents. In order to investigate the impact of metal centres on anticancer activity, Rh and Ir analogues of the most promising and orally active compound plecstatin (9) were prepared. Within the same group, the lighter metal fragments ruthenium and rhodium complexes showed increased cytotoxicity as compared to their respective heavier congener i.e. osmium and iridium. However, changing the halido leaving group resulted in slight decrease in activity with exception of ruthenium-bromido 17 and osmium-iodido 20 complexes in H460 cancer cell line. To further explore the carbonic anhydrase as another potential target for these compounds, PCA was functionalized with sulfonamide group and convert into RuII and OsII(η6-p-cymene) complexes. The presence of the sulfonamide motif in many organic drugs and metal complexes endowed these agents with interesting biological properties and may result in the latter case in multitargeting agents. The compounds were characterized with standard methods and the in vitro anticancer activity data was compared with studies on the hydrolytic stability of the complexes and their reactivity to small biomolecules. A molecular modelling study against carbonic anhydrase II revealed plausible binding modes of the complexes in the catalytic pocket. In a multitargeting approach, by incorporating several bioactive components – a metal centre, a pyridinecarbothioamide and a hydroxamic acid – in a novel pharmacophore, highly cytotoxic functionalized PCAs and their organometallic compounds were obtained. The PCA ligand 31 bearing the vorinostat (SAHA) pharmacophore and their respective organoruthenium, osmium, rhodium and iridium complexes 38–41 displayed potent cytotoxicity but these results showed slight correlation towards HDACi studies. In HDAC inhibition assay against HDAC1, HDAC6 and HDAC8, the PCA-SAHA derivative 31 and its organometallic compounds 38–41 showed inhibitory activity in nanomolar range and some derivatives were more potent inhibitors than the approved drug SAHA. The HDACi mechanism further confirmed by dynamic simulation where compound 31 and its enantiomeric complexes 39 and 18 40 chelated with Zn2+ ion of HDAC8 and HDAC6 and formed several interactions within their binding pocket. Overall, this doctoral thesis comprises of seven new ligands (6, 7, 23, 28–31) and twenty six novel organometallic complexes(10–18, 20–22, 24–27, 32–41), while single crystals of four ligands (3, 6, 23, 28) and seven complexes (12, 13, 17, 18, 20, 27neutral, 33) are reported.