Concavity Solutions of Second-Order Differential Equations

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1 Proceedigs of the Paista Academy of Scieces 5 (3): 4 45 (4) Copyright Paista Academy of Scieces ISSN: (prit), (olie) Paista Academy of Scieces Research Article Cocavity Solutios of Secod-Order Differetial Equatios Ibtisam Aldawish * ad Maslia Darus School of Mathematical Scieces, Faculty of Sciece ad Techology, Uiversiti Kebagsaa Malaysia, Bagi 436, Selagor Darul Ehsa, Malaysia Abstract: I this article, we cosider varieties of secod-order liear differetial equatios i the uit dis We show that the solutios of the secod-order liear differetial equatios are cocave uivalet fuctios uder some coditios Keywords: Aalytic fuctio, differetial equatio, cocave fuctio, uivalet fuctio AMS Mathematics Subject Classificatio: 3C45 INTRODUCTION Let A deote the class of fuctios ormalized by f ( z) = z a z, ( z ), () which are aalytic i the ope uit dis = { z : z < } o the complex plae For fuctios f A with f ( z ) ( z ), we defie the Schwarzia derivative of f by f ( z) f ( z) S( f, z),( f A; f ( z), z ) f ( z) f ( z) Let B deote the class of bouded fuctios q ( z ) q z q z aalytic i the uit dis, for which q( z ) K If g ( z ) B, the by usig the Schwarz lemma [8], the fuctio q ( z ) defied by ( ) z q z z g( t) t dt is also i B Thus, i terms of derivatives, we have ( ) ( ) < ( ), ( ) q z zq z K q z K z () If we let ( u, v ) u v We ca write () as ( q( z ), zq( z )) < K q( z ) K (3) Saitoh [] ad Millar [7] showed that (3) holds true for fuctios ( u, v ) i the class H give by Defiitio below Defiitio (see [7]) Let H be the set of complex fuctios ( u, v ) satisfyig the followig coditios: i ( u, v ) is cotiuous i a domai ; ii (,) ad (,) K ; i i iii ( Ke, Te ) K whe ( Ke, Te ), is real ad T K Defiitio (see [6]) Let H with correspodig domai We deote by B ( ) those fuctios q ( z ) q z q z which are aalytic i satisfyig : i ( q( z), zq( z)), i i Received, February 4; Accepted, August 4 *Correspodig author: Maslia Darus; maslia@umedumy

2 4 Ibtisam Aldawish ad Maslia Darus ii ( q( z), zq( z)) K ( z ) May other authors also studied the geometric properties solutios of a class of secod-order liear differetial equatios, for example oe ca refer to [, 4, 6, 7,,, ] We ow state the followig result due to Miller [7] Theorem 3 (Miller [7]) Let pz ( ) be a aalytic fuctio i the uit dis with zp( z ) < Let vz ( ), z, be the uique solutio of v( z) pz ( ) vz ( ), with v() ad v() The, zv ( z ) < v( z) ad v( z ) is a starlie coformal map of the uit dis Theorem 3 is related rather closely to some earlier results of Robertso [] ad Nehari [8], which we recall Theorem 4 ad Theorem 5, respectivelyas follows: Theorem 4 (Robertso []) Let zp( z ) be a aalytic fuctio i ad z pz ( ) z ( z ) The, the uique 4 solutio v v( z) of the followig iitial-value problem: v( z) pz ( ) vz ( ) ( v(), v() ) is uivalet ad starlie i The costat /4 is the best possible oe Theorem 5 (Nehari [8]) If f ( z) A ad it satisfies S( f, z) ( z ), the f ( z ) is uivalet The ext theorems, which are due to Saitoh [,] ad Owa et al [9], ivolve several geometric properties of the solutios of the secod-order liear differetial equatios Theorem 6 (Saitoh []) Let az ( ) ad bz ( ) be aalytic i with z bz ( ) a( z) [ az ( )] 4 ad az ( ) Let vz ( )( z ) be the solutio of the followig secod order liear differetial equatio v() z a() z v() z b() z v() z, v(), v() The, v( z) is starlie i Theorem 7 (Owa et al [9]) Let the fuctio az ( ) ad bz ( ) be aalytic i with za( z) Kad z bz ( ) a( z) [ az ( )] K Also, let ( ) 4 v z deote the solutio of the iitial- value problem equatio: v( z) a( z) v( z) b( z) v( z), v(), v() The, zv() z K { za( z)} vz () K { za( z)}, (z ; K ) Theorem 8 (Saitoh []) Let p ( z ) be the ocostat polyomial of degree with p ( z) K (z ; K ) Let v( z ) be the solutio of the iitial-value problem: v( z) p ( z) v( z), v()=; v () The, we have zv( z) K K ( z ) vz ( ) The followig theorem was proved by Abubaer ad Darus [] usig the third-order liear differetial equatio Theorem 9 (Abubaer & Darus []) Let Q( z ) b z be aalytic i with b K (z ; K ), ad let v( z ) deote the solutio of the iitial-value problem v( z) Qz ( ) v( z), z The, zv( z) K K ( z; K ) v( z)

3 Cocavity Solutios of Secod-Order Differetial Equatios 43 Next, we state the family of cocave fuctios which is our mai focus here A fuctio f : is said to belog to the family C ( ) if f satisfies the followig coditios: i f is aalytic i with the stadard ormalizatio f () f '() I additio it satisfies f () ii f maps coformally oto a set whose complemet with respect to is covex iii The opeig agle of f ( ) at is less tha or equal, (, ] The class C ( ) is referred to as the class of cocave uivalet fuctios ad for a detailed discussio about cocave fuctios, we refer to [, 3, 5] We recall the aalytic characterizatio for fuctios f i C ( ), (,]: f C( ) if ad oly if P ( z), z, where f z f ( z ) pf ( z ) z z f ( z) Before we establish our mai results, we eed to idicate to the followig theorems to prove our results Theorem (see []) For ay H, B ( ) B, ( H ; K ) Theorem leads us to immediately to the followig result, which was also give by Saitoh [] Theorem (see []) Let H ad bz ( ) be a aalytic fuctio i with bz ( ) K If the differetial equatio (, zq( z)) bz ( ), q(), q() has a solutio q( z) aalytic i, the K The objective of the preset paper is to ivestigate the cocavity of solutios of the secod-order liear differetial equatios MAIN RESULTS We derive the followig results by employig Theorem First, we cocetrate o the cocavity of the solutio of the followig iitialvalue problem: q''( z) az ( ) q( z) bz ( ) (4) Theorem Let az ( ), bz ( ) be aalytic fuctios i such that z b( z) K, ( z ; K ) (5) Let, z be the solutio of the iitial value problem (4) i The, z z K, where (, ] (6) Proof We recall f C ( ) if ad oly if P ( z) i, where f z zg '( z ) p ( z) f z g( z) with g ( z ) zf '( z ) We ote that p is aalytic i with p () If we set rz ( ) ( z ) (7) the, rz ( ) is aalytic i, r () ad (4) becomes r( z ) ( za( z )) r( z ) zr ( z ) z b ( z ) (8) Thus (8) ca be rewritte as ( r( z ), zr( z )) z b( z ), where ( s, t ) s ( za( z )) s t Sice i (,) s t is cotiuous i a domai ;

4 44 Ibtisam Aldawish ad Maslia Darus ii (,) ad (,) K; i i iii For ( Ke, Te ), is real ad i i i T K, ( Ke, Te ) K e K T T K We coclude that ( s, t) H From the hypothesis (5) ad by employig Theorem, we obtai that rz ( ) K, K Combie this with (7) we have zq( z ) K, K q( z) This leads to the followig relatios z K z z z z K z We fid that z K K z ad z K K z We ca simplify the last expressios ad obtai (6) This completes the proof of the theorem If we tae K i Theorem, the we deduce the followig corollary Corollary Let az ( ), bz ( ) be aalytic fuctios i such that z b( z), ( z ; (, ]) Let q( z ) be the solutio of the iitial value problem (4) The, C ( ) Example 3 Let az ( ) ad bz ( ) i Corollary The, for z ad, the solutio of the followig iitial-value problem : q''( z), q(), q() is give by si z C() We ext show that the followig differetial equatio q''( z) M( zqz ) ( ) (9) has a solutio, which is cocave uivalet i Theorem 4 Let M ( z ) be aalytic fuctios i such that z M ( z) K ( z, K ) () Let, z D be the solutio of the iitial value problem (9) The, K z z K, where (, ] () Proof If we put rz ( ) ( z ), () we see that rz ( ) is aalytic i, r () ad (9) becomes r( z ) r( z ) zr ( z ) z M ( z ) (3) We ca write this equality as ( r( z ), zr( z )) z M ( z ), where ( s, t) s s t It is easy to chec that the coditios of Defiitio are satisfied Therefore from () ad i order to apply Theorem, we obtai rz ( ) K, K, which implies that zq '( z ) K, K

5 Cocavity Solutios of Secod-Order Differetial Equatios 45 Hece we coclude that z K z q( z) K ( z ; K ; (,]) Thus, the proof is complete Next we obtai the Corollary by followig substitutig K i Theorem 4 Corollary 5 Let M ( z ) be aalytic fuctios i such that z M ( z ), ( z ; (,]) Let q ( z ) be the solutio of the iitial value problem (9) The, q( z ) C ( ) 3 CONCLUSIONS The varieties of secod-order liear differetial equatios i the uit dis are discussed Moreover, we showed that the solutios of the secod-order liear differetial equatios are cocave uivalet fuctios uder some coditios 4 ACKNOWLMEEDGNTS The wor reported i the article was supported by UKM's grat: AP-3-9 ad DIP-3-5 REFERENCES Abubaer, AA & M Darus Geometric properties solutios of a class of third-order liear differetial equatios ISRN Applied Mathematics, Article ID 49853, 5 pp () Avhadiev, FG, C Pommeree & K-J Wirths Sharp iequalities for the coefficiet of cocave schlicht fuctios Commetarii Mathematici Helvetici 8: 8-87 (6) 3 Avhadiev, FG & K-J Wirths Cocave schlicht fuctios with bouded opeig agle at ifiity Lobachevsii Joural of Mathematics 7: 3- (5) 4 Boyce, WE Elemetary differetial equatios Joh Willey & Sos, pp (977) 5 Cruz, L & C Pommeree O cocave uivalet fuctios Complex Variables ad Elliptic Equatios 5: (7) 6 Miller, SS & PT Mocau Secod-order differetial iequalities i the complex plae Joural of Mathematical Aalysis ad Applicatios 65: (978) 7 Miller, SS A class of differetial iequalities implyig boudedess Illiois Joural of Mathematics: (976) 8 Nehari, Z The Schwarzia derivative ad schlicht fuctios Bulleti of the America Mathematical Society 55 : (949) 9 Owa, S, H Saitoh, HM Srivastava & R Yamaawa Geometric properties of solutios of a class of differetial equatios Computers & Mathematics with Applicatios 47: (4) Robertso, MS Schlicht solutios of w '' pw Trasactios of the America Mathematical Society 76: 54-74(954) Saitoh, H Uivalece ad starlieess of solutios of w '' aw bw Aales Uiversitatis Mariae Curie Sodowsa Sectio A 53 : 9-6(999) Saitoh, H Geometric properties of solutios of a class of ordiary liear differetial equatios Applied Mathematics ad Computatio 87: (7)