Applied Mathematics Letters

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1 Applied Mathematics Letters 4 (011) Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: On the properties o a class o log-biharmonic unctions S.A. Khuri Department o Mathematics and Statistics, American University o Sharjah, United Arab Emirates a r t i c l e i n o a b s t r a c t Article history: Received 16 June 010 Received in revised orm 7 January 011 Accepted 7 January 011 Keywords: Harmonic Biharmonic Log-biharmonic Univalent Starlike Convex The aim o this work is to introduce and investigate a new class o log-biharmonic unctions. Certain geometrically motivated properties and results concerning starlikeness, convexity and univalence o elements within this class versus the corresponding harmonic unctions are obtained and discussed. In particular, we consider the oodman Sa conjecture and prove that the conjecture is true or the logarithms o unctions belonging to this class. 011 Elsevier Ltd. All rights reserved. 1. Introduction Complex-valued harmonic unctions that are univalent and sense preserving in the unit disk U can be written in the orm g + h, where h and g are analytic in U. A continuous complex-valued unction F u + iv in a domain D C is biharmonic i the Laplacian o F is harmonic, that is F is harmonic in D i F satisies the biharmonic equation ( F) 0, where 4. The class o biharmonic unctions includes the class o harmonic unctions and is a subclass o the class o polyharmonic unctions. A continuous complex-valued unction F u + iv in a domain D C is log-biharmonic i log F is biharmonic, that is the Laplacian o log F is harmonic. A unction is said to be log-harmonic in D i there is an analytic unction a and is a solution o the nonlinear elliptic partial dierential equation a. It has been shown that i is a nonvanishing log-harmonic mapping, then can be expressed as kl where k and l are analytic unctions in D. It is worth noting that in the latter case the Laplacian o the logarithm o the nonvanishing log-harmonic mapping is ero, that is (log ) 0. A harmonic unction F is locally univalent i the Jacobian o F, J F, J F F F 0. A unction F is orientation preserving i J F F F > 0. We say that a univalent biharmonic (harmonic) unction F, with F(0) 0, is starlike i the curve F(re it ) is starlike with respect to the origin or each 0 < r < 1. In other words, F is starlike i arg F(reit ) Re F F > 0 or 0. F address: skhoury@aus.edu /$ see ront matter 011 Elsevier Ltd. All rights reserved. doi: /j.aml

2 S.A. Khuri / Applied Mathematics Letters 4 (011) A univalent biharmonic (harmonic) unction, F, with F(0) 0 and F(reit ) 0 whenever 0 < r < 1, is said to be convex i the curve F(re it ) is convex or each 0 < r < 1. In other words, F is convex i arg F(reit ) > 0 or 0. The biharmonic equation arises in physical applications including linear elasticity theory and luid low. The biharmonic unctions, which are closely associated with the biharmonic unctions, appear in Stokes low problems as well as in radar imaging problems. There are several problems involving Stokes low which arise in engineering and biological transport phenomena. For the various applications o the biharmonic unctions see [1 3] and the reerences within. Recently, biharmonic and log-biharmonic unctions have been studied in a number o papers; see or example [1,,4,5]. For more details on harmonic mappings and the various deinitions introduced see [6 8]. The purpose o this work is to study a class o log-biharmonic unctions. Some geometrical properties related to starlikeness, convexity and univalence are examined. Further, we show that the oodman Sa conjecture (see [9]) is valid or the logarithm o unctions belonging to this class.. Properties o the class LBH In this work, we will consider the ollowing class o unctions: LBH {F : F λ 1 +λ, where is a nonvanishing log-harmonic mapping in the unit disk U and (0) 1, and are nonvanishing analytic unctions in U, λ 1 and λ (λ + 1 λ 0) are constants}. It will be shown that this elements in this class are log-biharmonic unctions; some geometrical properties related to starlikeness, convexity and univalence or elements in LBH versus the corresponding harmonic unctions and/or logharmonic unctions are derived. First, deine the linear operator L by L. The deinition leads to the ollowing two properties: L[α + βg] αl[ ] + βl[g], L[g] L[g] + gl[ ], where, g are C 1 unctions and α, β are complex constants. Theorem 1. For any F LBH, F is log-biharmonic. Proo. Let F λ 1 +λ LBH. Taking the logarithm o both sides, we have log F log + log + (λ 1 + λ ) log. Upon dierentiating both sides with respect to and, respectively, we get (log F) + λ 1 log + (λ 1 + λ )(log ), (log F) h + λ 1 log + (λ 1 + λ )(log ). From the deinition o the class LBH it is required that is log-harmonic, which means that (log ) (log ) 0. Dierentiating the latter two equations we have (log F) λ 1 log + λ 1 (log ) + λ 1 (log ), (log F) λ 1 log + λ 1 (log ) + λ 1 (log ). We note that (log F) (log F). Further dierentiation leads to (log F) λ 1 (log ) + λ 1 (log ) + λ 1 (log ). This latter equation and the act that (log ) 0 yields that (log F) 0. This means that F is log-biharmonic. Theorem. Let F λ 1 +λ LBH. Then: a. L[log F] L[log ] + L[log ] + (λ 1 + λ )L[log ], b. L n [log F] L n [log ] + L n [log ] + (λ 1 + λ )L n [log ], where n is an integer.

3 1144 S.A. Khuri / Applied Mathematics Letters 4 (011) Proo. We have log F log + log + (λ 1 + λ ) log. Simple calculation yields L[(λ 1 + λ )] 0. Further, using the product rule property o the operator L, we have L[(λ 1 + λ ) log ] (λ 1 + λ )L[log ] + log L[(λ 1 + λ )] (λ 1 + λ )L[log ]. Thereore, rom the linearity o the operator L we have L[log F] L[log ] + L[log ] + L[(λ 1 + λ ) log ] L[log ] + L[log ] + (λ 1 + λ )L[log ]. The proo o part (b) ollows rom (a) and induction. Corollary 1. Let F λ 1 +λ LBH. Then L n [log F] L[log F] Ln [log ] L[log ], n. Proo. From part (b) o Theorem we have L n [log F] (λ 1 + λ )L n [log ]. Upon dividing both sides o the last equation by L[log F] and using part (a) o Theorem, the results ollows. Theorem 3. Let F λ 1 +λ LBH. Assume that is starlike, λ1 + λ > 0 and Re( ) > Re( h ); then F is starlike. Proo. From the deinition o the operator L it ollows that or L[log ] L[], Re(L[log ]) Re > 0, which means that is starlike i and only i Re(L[log ]) > 0. From Theorem part (a) and the deinition o the operator L we have L[log F] L[log ] + L[log ] + (λ 1 + λ )L[log ] h + (λ 1 + λ )L[log ]. iven that Re( ) > Re( h ) and λ 1 + λ > 0, we get Re(L[log F]) Re h > (λ 1 + λ )Re(L[log ]) > 0. + (λ 1 + λ )Re(L[log ]) Since Re(L[log F]) Re( F F F ), it ollows that F is starlike. Theorem 4. Let F λ 1 +λ LBH. Then the Jacobian o log F, Jlog F, is given by J log F J F F h + (λ 1 + λ ) λ 1 log Re(L[log(log )]) + (λ 1 + λ )J log + Re h + λ 1 Re{log L[log( )]}.

4 S.A. Khuri / Applied Mathematics Letters 4 (011) Proo. Taking the logarithm o F λ 1 +λ then dierentiating both sides with respect to and, respectively, yields F F + λ 1 log + (λ 1 + λ ), (1) F F h + λ 1 log + (λ 1 + λ ). () Hence we have [ F F + λ 1 log + (λ 1 + λ ) ] + λ 1log + (λ 1 + λ ) + λ 1 (λ 1 + λ ) log + λ 1(λ 1 + λ ) log + λ 1 log + (λ 1 + λ ) + Re λ 1 log + (λ 1 + λ ), (3) [ F h F + λ 1 log + (λ 1 + λ ) ] h + λ 1log + (λ 1 + λ ) h + λ 1 (λ 1 + λ ) log + λ 1(λ 1 + λ ) log + λ 1 log + (λ 1 + λ ) h + Re λ 1 log + (λ 1 + λ ). (4) J log F J F F F F F h + λ 1 (λ 1 + λ ) log + log + (λ 1 + λ ) + (λ 1 + λ )Re h + λ 1 Re log h h + λ 1 (λ 1 + λ 1 ) log Re log + (λ 1 + λ ) + (λ 1 + λ )Re h + λ 1 Re log h h + (λ 1 + λ ) λ 1 log Re(L[log(log )]) + (λ 1 + λ )J log + Re h + λ 1 Re{log L[log( )]}. (5) Corollary. Let F λ 1 +λ LBH. I J log F (λ 1 + λ ) Proo. Since h, then the Jacobian o log F, J log F, is given by [ λ 1 log Re(L[log(log )]) + (λ 1 + λ )J log + Re h, it ollows that h L[log( )] h 0. and also L[log ] ].

5 1146 S.A. Khuri / Applied Mathematics Letters 4 (011) Further we have Re h Re h Re Re L[log ]. The result now ollows rom Theorem 4. Corollary 3. Assume the unction log is starlike and orientation preserving, Re{L[log ]} > 0, λ 1, λ > 0; then log F is orientation preserving and consequently locally univalent. h > 0 and Proo. Re > 0 is real; hence h L[log ] h h Re(L[log ]) > 0. From the proo o Theorem 3, log being starlike implies Re(L[log(log )]) > 0. Further, log being orientation preserving yields J log > 0. It ollows rom Corollary that J log F > 0, that is log F is orientation preserving and hence J log F 0, that is log F is locally univalent. Theorem 5. Let F λ 1 +λ LBH. Then a. i log F(reit ) (log ) (log ) + (λ 1 + λ )[(log ) (log ) ]. b. log F(re it ) (log ) +(log ) + (log ) + (log ) +(λ 1 +λ )[(log ) +(log ) + (log ) + (log ) ]. Proo. We have log F log + log + (λ 1 + λ ) log. From Theorem 1 we obtain (log F) (log ) + λ 1 log + (λ 1 + λ )(log ), (log F) (log ) + λ 1 log + (λ 1 + λ )(log ), (log F) λ 1 log + λ 1 (log ) + λ 1 (log ). Thereore we get log F(re it ) i(log F) i(log F) i(log ) i(log ) + i(λ 1 + λ )[(log ) (log ) ], and hence part (a) o the theorem ollows. Further, we have (log F) + (log F) (log F) (log ) + (log ) + ( λ 1 + λ )[(log ) + (log ) ]. Upon dierentiation we also have (log F) (log ) + λ 1 (log ) + (λ 1 + λ )(log ), (log F) (log ) + λ 1 (log ) + (λ 1 + λ )(log ), and hence we get (log F) + (log F) (log ) + (log ) + λ 1 [(log ) + (log ) ] + (λ 1 + λ )[ (log ) + (log ) ]. As a result we have (log F) + (log F) (log F) + (log F) + (log F) (log ) + (log ) + (log ) + (log ) + (λ 1 + λ )[(log ) + (log ) + (log ) + (log ) ].

6 S.A. Khuri / Applied Mathematics Letters 4 (011) But log F(re it ) [i(log F) i(log F) ] (log F) (log F) + (log F) (log F) (log F), and consequently part (b) o the theorem ollows. Corollary 4. Let F λ 1 +λ LBH with and constant unctions. Then arg log F(reit ) arg log (reit ). Proo. and are constant unctions; hence and (log ) (log ) 0, (log ) + (log ) + (log ) + (log ) 0. It ollows rom Theorem 5 that arg log F(reit ) log F(re it ) Im log F(re it ) (log F) + (log F) (log F) + (log F) + (log F) Re (log F) (log F) (log ) + (log ) + (log ) + (log ) Re (log ) (log ) arg log (reit ). In the subsequent theorem we consider the oodman Sa conjecture and prove that it is valid or the logarithms o unctions belonging to the class LBH. Theorem 6. For any non-constant F LBH, log F sends the subdisk < r onto a convex region or r 1, but onto a non-convex region or any 1 < r < 1. Proo. F LBH; hence it is given by F λ 1 +λ, where is log-harmonic. From the deinition o convexity we have log is convex arg log (reit ) > 0. Since log is harmonic, and i we urther assume that it is convex in 0 < r r 0 1, then the conclusion o the ollowing theorem proved by Ruscheweyh and Salinas [9, Theorem 1] holds (which is basically the oodman Sa conjecture). I K H (φ), 0 < r r 0 1, then (r) K H (φ), where K H denotes the class o all complex-valued harmonic univalent unctions on the unit disk D with (D) convex in the direction e iφ. By Corollary 4 we have arg log F(reit ) arg log (reit ) > 0. This means that F is also convex and hence the proo o the theorem ollows. Reerences [1] Z. Abdulhadi, Y. Muhanna, S. Khuri, Logharmonicity and logbiharmonicity properties o a nonlinear complex operator, International Journal o Nonlinear Operators Theory and Applications (1) (007) [] Z. Abdulhadi, Y. Muhanna, S. Khuri, On some properties o solutions o the biharmonic equation, Applied Mathematics and Computation 177 (1) (006) [3] S.A. Khuri, Biorthogonal series solution o Stokes low problems in sectorial regions, SIAM Journal on Applied Mathematics 56 (1996) [4] Z. Abdulhadi, Y. Muhanna, S. Khuri, On univalent solutions o the biharmonic equation, Journal o Inequalities and Applications (5) (005)

7 1148 S.A. Khuri / Applied Mathematics Letters 4 (011) [5] Z. Abdulhadi, Y. Muhanna, S. Khuri, On the univalence o the log-biharmonic mappings, Journal o Mathematical Analysis and Applications 89 () (004) [6] J.. Clunie, T. Sheil-Small, Harmonic univalent unctions, Annales Academiae Scientiarum Fennicae, Series AI 9 (1984) 3 5. [7] P. Duren, Univalent Functions, Springer-Verlag, New York, [8] P. Duren, Harmonic Mappings in the Plane, Cambridge University Press, 004. [9] St. Ruscheweyh, L. Salinas, On the preservation o direction-convexity and the oodman Sa conjecture, Annales Academiæ Scientiarium Fennicæ Series A I 14 (1989)