LAZHAR S INEQUALITIES AND THE S-CONVEX PHENOMENON. I.M.R. Pinheiro

Transcription

1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume , 57 6 LAZHAR S INEQUALITIES AND THE S-CONVEX PHENOMENON IMR Piheiro Received December 007 Abstract I this urther little article, we simply exted Lazhar s work o iequalities or covex uctios to those a little bit beyod: S-covex uctios Itroductio We seem to have developed the precursor, ad so hoorable, work o Proessors Hudzik ad Maligrada to a palatable level o suitability, or applicatios i diverse areas, by makig their theory more oudatioal i the pure scope o the Sciece I this urther work, we wish to exted Lazhar s work to S-covexity uctios V I Proessor Lazhar has made use, as see o 3, o the sources,, 6 We obviously simply trusted Proessor Lazhar s citatios, reereed by the editorial board o JIPAM Little by little, the use o S-covexity is prove By our extesios o results ad oudatioal works, we have developed may tools that may be used i Optimizatio whe dealig with uctios that almost look like covex uctios but are ot By splittig the domai o the uctio ito itervals, oe may make the whole uctio passive o work i Optimizatio with little eort I the ext sectio, the set o symbols, as well as the deiitios here used are explaied i detail Sectio 3 will brig the results exposed by Lazhar i his precursor work Sectio 4 brigs our ew theorems, results derived rom the extesio o Lazhar s theorems to S-covexity, alog with their proos Deiitios We use the symbols deied i 5: K s or the class o S-covex uctios i the irst sese, some S; K s or the class o S-covex uctios i the secod sese, some S; K 0 or the class o covex uctios; s or the costat S, 0 < S <, used i the irst deiitio o S-covexity; s or the costat S, 0 < S <, used i the secod deiitio o S-covexity We use the deiitios preseted i 5: 99 Mathematics Subject Classiicatio Primary: 6D0; Secodary: 6D5 Key words ad phrases: S-covexity, covex, S-covex, uctio, iequality, extesio, bouds, improvemet, reiemet

2 58 IMR PINHEIRO Deiitio A uctio : X > R, cotiuous see or argumetatio, is said to be s -covex i the iequality λx + λ s s y λ s x + λ s y holds λ 0,, x, y X such that X R + Deiitio is called s -covex, s, i the graph lies below a bet chord L betwee ay two poits, that is, or every compact iterval J I, with boudary J, it is true that sup J L sup J L Deiitio 3 A uctio : X > R C is said to be s -covex i the iequality λx + λy λ s x + λ s y holds λ 0,, x, y X such that X R + 3 Lazhar s Precursor Theorems Theorem 3 I is a covex uctio ad x, x,, x lie i its domai, N, >, the : x + x + + x x i x + x x + x x + x + Theorem 3 I is a covex uctio ad a,, a lie i its domai, N, >, the : where a = a++a b + b a + a a, ad b i = a ai, i =,, 4 Our Theorems: Extesios o Lazhar s Work to S-covex Fuctios As a coclusio, or this oe more precursor paper, we metio our ow results, all based o Lazhar s previous developmets Theorem 4 I is a S -covex, o-egative, uctio ad x, x,, x lie i its domai, the x + + x x i x + x s + s x + x s x + x s We have added the iormatio, which we believe to be essetial, based o well-posedess theory or Philosophy, to the theorem I the idex is ot atural ad does ot start i, we do get problems We have added the iormatio, which we believe to be essetial, based o well-posedess theory or Philosophy, to the theorem I the idex is ot atural ad does ot start i, we do get problems

3 LAZHAR S INEQUALITIES AND THE S-CONVEX PHENOMENON 59 Proo Usig the coditio o S -covexity, with t =, we obtai: s x + x x + x x + x + However, s s x x + x x i = x i x i = x i s x i, x i Replacig x i with its equivalet expressio, as above, oe gets: x + x x + x x + x + s x i x i With the subsequet applicatio o the coditio o S -covexity, oe gets: x + x s + s x + x s x i s s x + x x i s Theorem 4 I is a S covex, o-egative, uctio ad x, x,, x lie i its domai, the: x + + x x i s s x + x x + x x + x s + Proo Usig the coditio o S -covexity, with t =, we obtai: x + x x + x x + x + However, s x x + x x i = s s x i s x i = s s x i x i, s x i

4 60 IMR PINHEIRO Replacig x i with its equivalet expressio, as above, oe gets: x + x x + x x + x + s s s x i s x i With the subsequet applicatio o the coditio o S -covexity, oe gets: x + x x + x x + x + s s s x i x i Remark Cosiderig the exteded theorem or Ks ad = 3, we get: x + x + x 3 x x x 3 3 x + x s x + x 3 s 3 s x3 + x s Remark Cosiderig the exteded theorem or Ks or = 3, we get: x + x + x 3 x x x 3 3 s 3s x + x x + x 3 x3 + x 3 s Theorem 43 I is a S -covex, o-egative, uctio ad a,, a lie i its domai, the: s b + b s a + a a, where a = a++a s ad b i = s a a i, i =,, s Proo We ow use the exteded Jese s iequality 4: b + b a + a, ad so, b + b s s a + a s a + a, s s a + a or b + b s a + + a Applyig Jese s exteded iequality, we get:

5 LAZHAR S INEQUALITIES AND THE S-CONVEX PHENOMENON 6 b + b s s a + a s a + + a s Theorem 44 I is a S -covex, o-egative, uctio ad a,, a lie i its domai, the s b + b a + a s a, where a = a++a ad b i = a ai, i =,, Proo We ow use the exteded Jese s iequality 4: b + b s a + a, ad so, b + b s a + a s a + a, s a + a or b + b s s s a + + s a Applyig Jese s exteded iequality, we get:, b + b s a + a s s a + + a, Reereces DS Mitriovíc, JE Pećaric ad AM Fik, Classical ad New Iequalities i Aalysis, Kluwer Academic Publishers, Dordrecht, 993 Kira Kedlaya, A B A is less tha B, based o otes or the Math Olympiad Program MOP Versio 0, Lazhar Bougoa, New iequalities about covex uctios, JIPAM, V 7, I 4, Art 48, MR Piheiro, Jese s iequality i detail ad S-covexity, Submitted, 008 Olie preprit located at wwwgeocitiescom/msor iap or wwwscribdcom/illmrpiheiro 5 MR Piheiro, Explorig the cocept o S-covexity, Aequatioes Mathematicae, Acc 006, V 74, I3, 007, T Popoviciu, Sur certaies iégalitées qui caractériset les uctios covexes, A Sti Uiv Al I Cuza Iasi I-a, Mat N S, 965, 55 64

6 6 IMR PINHEIRO IMR Piheiro PO BOX 396 A Beckett st Melboure Victoria, 8006 AUSTRALIA mrpproessioal@yahoocom