Journal of Mathematical Analysis and Applications
|
|
- Maryann McGee
- 5 years ago
- Views:
Transcription
1 J. Math. Anal. Appl Contents lists available at ScienceDirect Journal o Mathematical Analysis and Applications Value sharing results or shits o meromorphic unctions Xudan Luo, Wei-Chuan Lin Department o Mathematics, Fujian Normal University, Fuzhou , Fujian Province, PR China article ino abstract Article history: Received 7 May 200 Available online 5 November 200 Submitted by Steven G. Krantz Keywords: Uniqueness Entire unction Dierence product In this paper, we deal with the value distribution o dierence products o entire unctions, and present some result on two dierence products o entire unctions sharing one value with the same multiplicities. The research indings also include an analogue or shit o a well-known conjecture by Brück. Our theorems improve the results o I. Laine and C.C. Yang [I. Laine, C.C. Yang, Value distribution o dierence polynomials, Proc. Japan Acad. Ser. A Math. Sci ], K. Liu and L.Z. Yang [K. Liu, L.Z. Yang, Value distribution o the dierence operator, Arch. Math ], and J. Heittokangas et al. [J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J.L. Zhang, Value sharing results or shits o meromorphic unction, and suicient conditions or periodicity, J. Math. Anal. Appl ]. Moreover, we show by illustrating a number o examples that our results are best possible in certain senses. Published by Elsevier Inc.. Introduction and main results In the paper, we assume all the unctions are nonconstant meromorphic unctions in the complex plane C. We shall use the ollowing standard notations o value distribution theory: T r,, mr,, Nr,, Nr,, Sr,,... See, e.g. [7,3]. We denote by Sr, any unction satisying Sr, = o { T r, }, as r +, possibly outside o a set with inite measure. We speciy the notion o small unctions as ollows: Given a meromorphic unction, the amily o all meromorphic unctions az such that T r, a = Sr, is denoted by S. For convenience, we also include all constant unctions in S. Moreover,Ŝ = S { }. I or some a C { },thezeroso a and g a coincide in locations and multiplicity, we say that and g share the value a CM. There has been an increasing interest in studying dierence equations and dierence product in the complex plane. Halburd and Korhonen [5] established a version o Nevanlinna theory based on dierence operators. Bergweiler and Langley [2] considered the value distribution o zeros o dierence operators that can be viewed as discrete analogues o zeros o z. Ishizaki and Yanagihara [9] developed a version o Wiman Valion theory or dierence equations o entire unctions o small growth. Growth estimates or the dierence analogue o the logarithmic derivative z+c z were given by Halburd and Korhonen [4] and Chiang and Feng [3] independently. Recently, Laine and Yang [0] investigated the value distribution o dierence products o entire unctions, and obtained the ollowing result. The project sponsored by SRF or ROCS, SEM, and the National Natural Science Foundation o China Grant No * Corresponding author. address: sxlwc936@jnu.edu.cn W.-C. Lin X/$ see ront matter Published by Elsevier Inc. doi:0.06/j.jmaa
2 442 X. Luo, W.-C. Lin / J. Math. Anal. Appl Theorem A. Let z be a transcendental entire unction o inite order, and c be a nonzero complex constant. Then or n 2, z n z + c assumes every nonzero value a C ininitely oten. The restriction in Theorem A to the inite order case is essential. As an example, take z = exp e z and e c = n, then is o ininite order such that z n z + c. Theorem A does not remain valid i n =. Indeed, take z = + e z,then z z + π i = e 2z. Aterwards, Liu and Yang [] improved Theorem A, and proved the ollowing result. Theorem B. Let z be a transcendental entire unction o inite order, and c be a nonzero complex constant. Then or n 2, z n z + c pz has ininitely many zeros, where Pz 0 is a polynomial in z. In this paper, we will establish an improvement o Theorem A and Theorem B, which is stated as ollows. Theorem. Let be a transcendental entire unction o inite order σ and c be a ixed nonzero complex constant, let Pz = a n z n + a n z n + +a z + a 0 be a nonzero polynomial, where a 0, a,...,a n 0 are complex constants, and m is the number o the distinct zeros o Pz. Then or n > m, P z + c = az has ininitely many solutions, where az S \{0}. We shall give a much simple proo o Theorem in Section 3, which is dierent rom Res. [0,]. Remark. The ollowing examples show that Theorem A, Theorem B and Theorem may ail to occur or meromorphic unctionsoiniteorder. Example. Let Pz = z 3 z 2 + 2, = tan z, c = π 2.WegetP z + π 2 = or this example. Example. Let Pz = z 2, z = 3tanz +, c = 2π 3.WegetP z + 2π 3 = 6 and Theorem ail to occur or this example. +. Clearly, Theorem ails to occur cos 4 z 8 8. Clearly, Theorem A, Theorem B cos 2 z Corresponding to the above result, we investigate the uniqueness o dierence products o entire unctions, and obtain the next result. For the sake o simplicity, we use the deinition as ollows. Deinition. Let Pz = a n z n + a n z n + +a z + a 0 a n 0, wedenoteγ 0 = m + 2m 2, where m is the number o the simple zero o Pz, and m 2 is the number o multiple zeros o Pz. Wedenoted = GCD{λ 0,λ,...,λ n }, where { i +, ai 0, λ i = n +, a i = 0, i = 0,, 2,...,n. Theorem 2. Let and g be transcendental entire unctions o inite order, c be a nonzero complex constant, Pz = a n z n +a n z n + +a z + a 0 be a nonzero polynomial, where a 0, a,...,a n 0 are complex constants, and let n > 2Γ 0 + be an integer. I P z + c and Pggz + c share CM, then one o the ollowing results holds: tg or a constant t such that t d =,wheredisdeinedabove; 2 and g satisy the algebraic equation R, g 0,whereRw, w 2 = Pw w z + c Pw 2 w 2 z + c; 3 z = e αz,gz = e βz,whereαz and βz are two polynomials, b is a constant satisying α + β banda 2 n en+b =. Remark 2. The ollowing example shows that the second case o Theorem 2 may occur. Let Pz = z 6 z + 6 z, z = sin z, gz = cos z and c = 2π. It is easy to see that n > 2Γ 0 + and P z + c Pggz + c, sop z + c and Pggz + c share CM. Clearly, we get tg or a constant t such that t m =, where m Z +, but and g satisy the algebraic equation R, g 0, where Rw, w 2 = Pw w z + c Pw 2 w 2 z + c. However, when Pz is a nonzero monomial, the second case o Theorem 2 may be deleted. Indeed, or instance, let Pz = z n, n > 2, then rom the algebraic equation R, g 0, we have n = g gz + c z + c.
3 X. Luo, W.-C. Lin / J. Math. Anal. Appl This relation and Lemma 3 yields that n T r, = S r,. g g It ollows that g is a constant, that is, tg or a constant t such that tn+ =. Thereore, we obtain the ollowing result. Corollary. Let and g be transcendental entire unctions o inite order, c be a nonzero complex constant. Suppose that n z + c and g n gz + c share CM, and n > 5 is an integer, then one o the ollowing results holds: tg or a constant t such that t n+ = ; 2 g t,wheret n+ =. We continue to our study in this paper by establishing shared value problems related to a meromorphic unction z and its shit z + c, where c C. Currently, J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. Zhang [8] obtained that i z is o inite order and shares two values CM and one value IM Ignoring multiplicities with its shit z + c, then is a periodic unction with period c. It is natural to investigate whether there exist uniqueness theorems under reducing the number o the shared small periodic unctions. However, the ollowing two counterexamples show that may not be a periodic unction i z and z + c only have two shared values. Example. Let = e z sin z +, c = 2π. Clearly, we get and z + c share CM, but is not periodic unctions. Example. Let = ez sin z +, c = 2π. sin z Clearly, we get and z + c share, CM, but is not periodic unctions. z However, rom the above examples, we ind that the two unctions satisy = τ or some constant τ.thesuch z+c result can be named as a shit analogue o Brück s conjecture []. The ollowing result is due to Heittokangas et al., see [8, Theorem ]. Theorem C. Let be a meromorphic unction o order o growth ρ := lim sup r log T r, < 2, log r and let c C.I z and z + C shared the values a C and CM, then z a z + c a = τ or some constant τ. Here, we also study the shit analogue o Brück s conjecture by relaxing the growth condition, and obtain the result as ollows. Theorem 3. Let be a nonconstant meromorphic unction o inite order, n 6 be an integer. I n and n z+c share az S \{0} and CM, then wz + c, or a constant w satisying w n =. 2. Some lemmas Next, or the proo o our theorems, we still need the ollowing lemmas. Lemma. See [2]. Let and g be two nonconstant meromorphic unction. I and g share CM, one o the ollowing three cases holds:
4 444 X. Luo, W.-C. Lin / J. Math. Anal. Appl T r, N 2 r, + N 2 r, g + N 2 r, + N 2r, + Sr, + Sr, g, the same inequality holding or T r, g; g 2 g; 3 g. Lemma 2. See [4,5]. Let z be a meromorphic unction o inite order σ, and let c be a ixed nonzero complex constant. Then or each ε > 0,wehave m r, z + c z + m r, z = O r σ +ε. z + c Lemma 3. See [3]. Let be a meromorphic unction o inite order σ,c 0 be ixed. Then or each ε > 0, wehave T r, z + c = T r, + O r σ +ε + Olog r. Lemma 4. See [6]. Let T : 0, + 0, + be a non-decreasing continuous unction, s > 0, α <,andletf R + be the set o all r such that T r αt r + s. I the logarithmic o F is ininite, then lim sup r log + T r, =. log r Lemma 5. Let be an entire unction o inite order σ, c be a ixed nonzero complex constant, and let Pz = a n z n + a n z n + + a z + a 0 be a nonzero polynomial, where a 0, a,...,a n 0 are complex constants. Then or each ε > 0, wehave T r, P z + c = T r, P z + O r σ +ε. Proo. Since has inite order σ, rom Lemma 2, we deduce that T r, P z + c = m r, P z + c m r, P z + c + m r, m r, P + O r σ +ε = T r, P + O r σ +ε. Similarly, we deduce T r, P = m r, P m r, P z + c + m r, z + c m r, P z + c + O r σ +ε = T r, P z + c + O r σ +ε. Thereore, T r, P z + c = T r, P z + Or σ +ε. Remark 3. Under the condition o Lemma 5, we have Sr, P z + c = Sr, and Sr, = Sr, P z + c. Remark 4. The ollowing example shows Lemma 5 may ail to occur or meromorphic unctions o inite order. Example. Let Pz = z n, n Z +, = z tan z, c = π 2. We get P z + π 2 = zn z + π 2 tann z and P z = z n+ tan n+ z. Clearly, Lemma 5 ails to occur or this example. Lemma 6. See [3]. Let and g be two nonconstant meromorphic unctions such that and g share, CM. I N 2 r, + N 2 r, + 2Nr, <λtr + Sr, g where λ<, Tr = max{t r,, T r, g}, Sr = 0{T r} r,r / E, and E has inite linear measure. Then gor g.
5 X. Luo, W.-C. Lin / J. Math. Anal. Appl Lemma 7. See [3]. Let be a nonconstant meromorphic unction, and let a 0, a,...,a n be inite complex numbers such that a n 0. Then T r, a n n + a n n + +a 0 = ntr, + Sr,. Lemma 8. Let be an meromorphic unction o inite order, c 0 be ixed. Then N r, N r, + Sr,, z + c N r, z + c Nr, + Sr,, N r, N r, + Sr,, z + c N r, z + c Nr, + Sr,, outside o a possible exceptional set with inite logarithmic measure. Proo. We will use the method o proo o Re. [8, Theorems 6, 7] to prove this lemma. By a simple geometric observation, we have N r, N r + c,. z + c Since the order o is inite, by Lemma 4, we obtain N r + c, N r, + Sr,, outside o a possible exceptional set with inite logarithmic measure. On the other hand, we have Nr, Nr + c,. Thereore, Nr + c, = Nr, + Sr,. From above, we get N r, z + c N Similarly, we obtain that r, + Sr,. N r, z + c Nr, + Sr,, N r, N r, + Sr,, z + c N r, z + c Nr, + Sr,, outside o a possible exceptional set with inite logarithmic measure. 3. Proo o Theorem Contrary to the assertion, suppose that P z + c = az has initely solutions, then by the Second Fundamental Theorem, Lemma 5 and Lemma 8, we have T r, P z + c N r, + N r, + Sr, P z + c P z + c az N r, + N r, + Sr, P z + c N r, + N r, + Sr, P m + T r, + Sr,. From Lemma 5, we get T r, P m + T r, + Sr,,
6 446 X. Luo, W.-C. Lin / J. Math. Anal. Appl i.e. n + T r, m + T r, + Sr,, which contradicts with n > m. Thus, we have completed the proo o Theorem. 4. Proo o Theorem 2 Let F = p z + c, G = pggz + c, thenf and G share CM. Applying Lemma to F and G, wegetthatoneo the ollowing three cases holds. Case. I T r, F N 2 r, F + N 2r, + Sr, F + Sr, G, by Lemma 5 and Lemma 8, we have G T r, F N 2 r, + N 2 r, + Sr, + Sr, g F G = N 2 r, + N 2 r, p z + c pggz + c N 2 r, + N 2 r, + N 2 r, p z + c pg Γ 0 T r, + Γ 0 T r, g + N r, + N r, Γ 0 T r, + Γ 0 T r, g + N r, z + c + N r, g Γ 0 + T r, + Γ 0 + T r, g + Sr, + Sr, g. From Lemma 5, we have + Sr, + Sr, g + N 2 r, gz + c + Sr, + Sr, g T r, P Γ 0 + T r, + Γ 0 + T r, g + Sr, + Sr, g. By Lemma 7, we deduce + Sr, + Sr, g gz + c + Sr, + Sr, g n + T r, Γ 0 + T r, + Γ 0 + T r, g + Sr, + Sr, g. 4. Similarly, we obtain n + T r, g Γ 0 + T r, + Γ 0 + T r, g + Sr, + Sr, g. 4.2 Combining 4. and 4.2, we have n + [ T r, + T r, g ] 2Γ [ T r, + T r, g ] + Sr, + Sr, g, which contradicts with n > 2Γ 0 +. Case 2. I F G, that is P z + c Pggz + c. 4.3 Set h =,ih is a constant, then substituting = gh into 4.3, we deduce that g gz + c [ a n g n h n+ + a n g n h n + +a 0 h ] 0, where a 0, a,...,a n 0 are complex constants. Since g is transcendental entire unction, hence gz + c 0. From above, we get a n g n h n+ + a n g n h n + +a 0 h We claim that h d =, where d is deined as in Deinition. Thus, tg or a constant t such that t d =. In act, we discuss the ollowing subcases. Subcase. Suppose that a n is the only nonzero coeicient. Since g is transcendental entire unction, we have h n+ =.
7 X. Luo, W.-C. Lin / J. Math. Anal. Appl Subcase 2. Suppose that a n is not the only nonzero coeicient. I h n+, by Lemma 7 and 4.4, wededucet r, g = Sr, g, which is a contradiction. Hence, h n+ =. According to the similar discussion, we obtain that h k+ = when a k 0 or some k = 0,...,n. Thereore, we get tg or a constant t such that t d =, where d = GCDλ 0,λ,...,λn. I h is not a constant,then we know by 4.3 that and g satisy the algebraic equation R, g 0, where Rw, w 2 = Pw w z + c Pw 2 w 2 z + c. Case 3. I FG, that is P z + cpggz + c. From the assumption that and g are two nonconstant entire unctions, we deduce by 4.5 that P 0, Pg 0. By Picard s theorem, we claim that P = a n a n, Pg = a n g a n, where a is a complex constant. Otherwise, the Picard s exceptional values are at least three, which is a contradiction. Hence, rom the assumption that and g be transcendental entire unctions o inite order, we obtain that z = e αz + a, gz = e βz + a, where αz and βz are two nonconstant polynomials. By 4.5, we also get z + c 0, gz + c 0. So a = 0, i.e. z = e αz, gz = e βz, Pz = a n z n, and a 2 n enαz+βz+αz+c+βz+c. Dierentiating this yields n α z + β z + α z + c + β z + c Let ωz = α z + β z, we deduce by 4.6 that nωz + ωz + c 0. Since ωz is a polynomial, we suppose degωz = m, and z,...,z m are the zeros o ωz. Thus,z + c,...,z m + c are also the zeros o ωz. Thereore, ω 0, α + β b, where b is a constant. From this we can easily obtain that z = e αz, gz = e βz, where αz and βz are two polynomials, b is a constant satisying α + β b and a 2 n en+b =. This completes the proo o Theorem Proo o Theorem Let F = n az, G = n z+c az,thenf and G share, CM. By Lemma 3, we have Sr, G = Sr,. Set F H = F F G GG. We distinguish two cases as ollows. 5. Case. I H 0, we get F B B G, where B is a nonzero constant. We claim that F G, that is, n n z + c, which implies that wz + c, or a constant w satisying w n =. In act, we discuss the ollowing subcases. Subcase. Suppose that Nr, Sr,, then there exists z which is not a zero or pole o az such that F and F z + c share CM, so F z = Gz = 0. We get rom 5. that B =, so F G. Subcase 2. Suppose that Nr, = Sr,. IB, then we have N r, = Nr, G = Sr, F. B Hence, T r, F Nr, F + N r, + N r, F F B which contradicts with n 6. Thereore, B =. Thus, F G. + Sr, F N r, + Sr,, z = 0. Since
8 448 X. Luo, W.-C. Lin / J. Math. Anal. Appl Case 2. I H 0, by Lemma 8 and 5., we deduce that Nr, H N r, + N r, + Sr, z + c 2N r, + Sr, 2T r, + Sr,. 5.2 Thereore, by a logarithmic derivative theorem and 5.2, we get T r, H 2T r, + Sr,. 5.3 Suppose that z 0 is a pole o with multiplicity p, then an elementary calculation gives that z 0 is the zero o H with multiplicity at least np. From this and 5.3, we have n Nr, N Hence, r, H Nr, 2 T r, + Sr,. n By 5.4 and Lemma 8, we obtain that N 2 r, + N 2 r, F G 2T r, + Sr,. + 2Nr, F 2N 4N r, + 2N r, z + c r, + 2Nr, + Sr, Set T r = max{t r, F, T r, G}=nTr, + Sr,. From 5.5, we have N 2 r, + N 2 r, + 2Nr, F n T r + Sr,, F G n + 2Nr, + Sr, T r, + Sr,. 5.5 n which contradicts with n 6. Using Lemma 6, we get F G or FG. I F G, that is, n n z + c, which implies that wz + c, or a constant w satisying w n =. I FG, that is F zf z + c a 2 z, which implies Nr, F = Nr, = Sr,. Since F 2 z = a2 zf z F z+c, mr, F = Sr,, F z F z+c = N r, a2 z F 2 zf z+c,wehave F z F z + c = Sr,. Hence, T r. = Sr,, which is a contradiction. This completes the proo o Theorem Discussion We irstly denote M by the set o meromorphic unctions in complex plane such that Nr, = Sr,. In irst section, we have discussed the value distribution o dierence products o entire unctions, and presented the examples to show that Theorem is not valid or meromorphic unctions. It remains an open question under what conditions Theorem holds or meromorphic unctions o inite order. Here, we show that this problem is valid or the meromorphic unction M with inite order. Theorem 4. Let M be a meromorphic unction o inite order σ and c be a ixed nonzero complex constant, let Pz = a n z n + a n z n + +a z + a 0 be a nonzero polynomial, where a 0, a,...,a n 0 are complex constants, and m is the number o the distinct zeros o Pz. Then or n > m, P z + c = az has ininitely many solutions, where az S \{0}. F z+c
9 X. Luo, W.-C. Lin / J. Math. Anal. Appl Acknowledgments We thank the reerees or reading the manuscript careully and making a number o valuable comments and suggestions. Reerences [] R. Brück, On entire unctions which share one value CM with their irst derivative, Results Math [2] W. Bergweiler, J.K. Langley, Zeros o diereces o meromorphic unctions, Math. Proc. Cambridge Philos. Soc [3] Y.M. Chiang, S.J. Feng, On the Nevanlinna characteristic z + η and dierence equations in the complex plane, Ramanujan J [4] R.G. Halburd, R. Korhonen, Dierence analogue o the lemma on the logarithmic derivative with application to dierence equations, J. Math. Anal. Appl [5] R.G. Halburd, R. Korhonen, Nevanlinna theory or the dierence operator, Ann. Acad. Sci. Fenn. Math [6] R.G. Halburd, R. Korhonen, Finite-order meromorphic solutions and the discrete Painleve equations, Proc. Lond. Math. Soc [7] W.K. Hayman, Meromorphic Functions, Clarendon Press, Oxord, 964. [8] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J.L. Zhang, Value sharing results or shits o meromorphic unction, and suicient conditions or periodicity, J. Math. Anal. Appl [9] K. Ishzaki, N. Yanagihara, Wiman Valiron method or dierence equations, Nagoya Math. J [0] I. Laine, C.C. Yang, Value distribution o dierence polynomials, Proc. Japan Acad. Ser. A Math. Sci [] K. Liu, L.Z. Yang, Value distribution o the dierence operator, Arch. Math [2] C.C. Yang, X.H. Hua, Uniqueness and value-sharing o meromorphic unctions, Ann. Acad. Sci. Fenn. Math [3] C.C. Yang, H.X. Yi, Uniqueness Theory o Meromorphic Functions, Math. Appl., vol. 557, Kluwer Academic Publishers Group, Dordrecht, 2003.
On Picard value problem of some difference polynomials
Arab J Math 018 7:7 37 https://doiorg/101007/s40065-017-0189-x Arabian Journal o Mathematics Zinelâabidine Latreuch Benharrat Belaïdi On Picard value problem o some dierence polynomials Received: 4 April
More information1 Introduction, Definitions and Results
Novi Sad J. Math. Vol. 46, No. 2, 2016, 33-44 VALUE DISTRIBUTION AND UNIQUENESS OF q-shift DIFFERENCE POLYNOMIALS 1 Pulak Sahoo 2 and Gurudas Biswas 3 Abstract In this paper, we deal with the distribution
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 352 2009) 739 748 Contents lists available at ScienceDirect Journal o Mathematical Analysis Applications www.elsevier.com/locate/jmaa The growth, oscillation ixed points o solutions
More informationWEIGHTED SHARING AND UNIQUENESS OF CERTAIN TYPE OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS
Palestine Journal of Mathematics Vol. 7(1)(2018), 121 130 Palestine Polytechnic University-PPU 2018 WEIGHTED SHARING AND UNIQUENESS OF CERTAIN TYPE OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS Pulak Sahoo and
More informationAnalog of Hayman Conjecture for linear difference polynomials
Analog of Hayman Conjecture for linear difference polynomials RAJ SHREE DHAR Higher Education Department, J and K Government Jammu and Kashmir,INDIA dhar.rajshree@jk.gov.in September 9, 2018 Abstract We
More informationResearch Article Uniqueness Theorems of Difference Operator on Entire Functions
Discrete Dynamics in Nature and Society Volume 203, Article ID 3649, 6 pages http://dx.doi.org/0.55/203/3649 Research Article Uniqueness Theorems of Difference Operator on Entire Functions Jie Ding School
More informationResearch Article Fixed Points of Difference Operator of Meromorphic Functions
e Scientiic World Journal, Article ID 03249, 4 pages http://dx.doi.org/0.55/204/03249 Research Article Fixed Points o Dierence Operator o Meromorphic Functions Zhaojun Wu and Hongyan Xu 2 School o Mathematics
More informationZeros and value sharing results for q-shifts difference and differential polynomials
Zeros and value sharing results for q-shifts difference and differential polynomials RAJ SHREE DHAR Higher Education Department, J and K Government Jammu and Kashmir,INDIA dhar.rajshree@jk.gov.in October
More informationUNIQUENESS OF ENTIRE OR MEROMORPHIC FUNCTIONS SHARING ONE VALUE OR A FUNCTION WITH FINITE WEIGHT
Volume 0 2009), Issue 3, Article 88, 4 pp. UNIQUENESS OF ENTIRE OR MEROMORPHIC FUNCTIONS SHARING ONE VALUE OR A FUNCTION WITH FINITE WEIGHT HONG-YAN XU AND TING-BIN CAO DEPARTMENT OF INFORMATICS AND ENGINEERING
More informationPROPERTIES OF SCHWARZIAN DIFFERENCE EQUATIONS
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 199, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PROPERTIES OF
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 355 2009 352 363 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Value sharing results for shifts of meromorphic
More informationDIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
Journal o Applied Analysis Vol. 14, No. 2 2008, pp. 259 271 DIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS B. BELAÏDI and A. EL FARISSI Received December 5, 2007 and,
More informationResearch Article Unicity of Entire Functions concerning Shifts and Difference Operators
Abstract and Applied Analysis Volume 204, Article ID 38090, 5 pages http://dx.doi.org/0.55/204/38090 Research Article Unicity of Entire Functions concerning Shifts and Difference Operators Dan Liu, Degui
More informationn(t,f) n(0,f) dt+n(0,f)logr t log + f(re iθ ) dθ.
Bull. Korean Math. Soc. 53 (2016), No. 1, pp. 29 38 http://dx.doi.org/10.4134/bkms.2016.53.1.029 VALUE DISTRIBUTION OF SOME q-difference POLYNOMIALS Na Xu and Chun-Ping Zhong Abstract. For a transcendental
More informationResearch Article Uniqueness Theorems on Difference Monomials of Entire Functions
Abstract and Applied Analysis Volume 202, Article ID 40735, 8 pages doi:0.55/202/40735 Research Article Uniqueness Theorems on Difference Monomials of Entire Functions Gang Wang, Deng-li Han, 2 and Zhi-Tao
More informationFixed points of the derivative and k-th power of solutions of complex linear differential equations in the unit disc
Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 48, -9; http://www.math.u-szeged.hu/ejqtde/ Fixed points of the derivative and -th power of solutions of complex linear differential
More informationOn the Uniqueness Results and Value Distribution of Meromorphic Mappings
mathematics Article On the Uniqueness Results and Value Distribution of Meromorphic Mappings Rahman Ullah, Xiao-Min Li, Faiz Faizullah 2, *, Hong-Xun Yi 3 and Riaz Ahmad Khan 4 School of Mathematical Sciences,
More informationSOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH EXCEPTIONALLY FEW ZEROS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 429 452 SOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH EXCEPTIONALLY FEW ZEROS Gary G. Gundersen, Enid M. Steinbart, and
More informationFIXED POINTS OF MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 3, 2006, 9 2 FIXED POINTS OF MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS Liu Ming-Sheng and Zhang Xiao-Mei South China Normal
More informationGrowth of Solutions of Second Order Complex Linear Differential Equations with Entire Coefficients
Filomat 32: (208), 275 284 https://doi.org/0.2298/fil80275l Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Growth of Solutions
More informationRELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 147 161. RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL
More informationSOME PROPERTIES OF MEROMORPHIC SOLUTIONS FOR q-difference EQUATIONS
Electronic Journal of Differential Equations, Vol. 207 207, No. 75, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOME PROPERTIES OF MEROMORPHIC SOLUTIONS FOR q-difference
More informationSlowly Changing Function Oriented Growth Analysis of Differential Monomials and Differential Polynomials
Slowly Chanin Function Oriented Growth Analysis o Dierential Monomials Dierential Polynomials SANJIB KUMAR DATTA Department o Mathematics University o kalyani Kalyani Dist-NadiaPIN- 7235 West Benal India
More informationWhen does a formal finite-difference expansion become real? 1
When does a formal finite-difference expansion become real? 1 Edmund Y. M. Chiang a Shaoji Feng b a The Hong Kong University of Science & Technology b Chinese Academy of Sciences Computational Methods
More informationNonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 20, Number 1, June 2016 Available online at http://acutm.math.ut.ee Nonhomogeneous linear differential polynomials generated by solutions
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics SOME NORMALITY CRITERIA INDRAJIT LAHIRI AND SHYAMALI DEWAN Department of Mathematics University of Kalyani West Bengal 741235, India. EMail: indrajit@cal2.vsnl.net.in
More informationHouston Journal of Mathematics. c 2008 University of Houston Volume 34, No. 4, 2008
Houston Journal of Mathematics c 2008 University of Houston Volume 34, No. 4, 2008 SHARING SET AND NORMAL FAMILIES OF ENTIRE FUNCTIONS AND THEIR DERIVATIVES FENG LÜ AND JUNFENG XU Communicated by Min Ru
More informationUNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING THREE VALUES
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 105 123. UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING THREE VALUES ARINDAM SARKAR 1 AND PAULOMI CHATTOPADHYAY 2 Abstract. We prove four uniqueness
More informationGrowth of meromorphic solutions of delay differential equations
Growth of meromorphic solutions of delay differential equations Rod Halburd and Risto Korhonen 2 Abstract Necessary conditions are obtained for certain types of rational delay differential equations to
More informationGrowth of solutions and oscillation of differential polynomials generated by some complex linear differential equations
Hokkaido Mathematical Journal Vol. 39 (2010) p. 127 138 Growth of solutions and oscillation of differential polynomials generated by some complex linear differential equations Benharrat Belaïdi and Abdallah
More informationON FACTORIZATIONS OF ENTIRE FUNCTIONS OF BOUNDED TYPE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 345 356 ON FACTORIZATIONS OF ENTIRE FUNCTIONS OF BOUNDED TYPE Liang-Wen Liao and Chung-Chun Yang Nanjing University, Department of Mathematics
More informationMEROMORPHIC FUNCTIONS AND ALSO THEIR FIRST TWO DERIVATIVES HAVE THE SAME ZEROS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 205 28 MEROMORPHIC FUNCTIONS AND ALSO THEIR FIRST TWO DERIVATIVES HAVE THE SAME ZEROS Lian-Zhong Yang Shandong University, School of Mathematics
More informationBANK-LAINE FUNCTIONS WITH SPARSE ZEROS
BANK-LAINE FUNCTIONS WITH SPARSE ZEROS J.K. LANGLEY Abstract. A Bank-Laine function is an entire function E satisfying E (z) = ± at every zero of E. We construct a Bank-Laine function of finite order with
More informationOn the value distribution of composite meromorphic functions
On the value distribution of composite meromorphic functions Walter Bergweiler and Chung-Chun Yang Dedicated to Professor Klaus Habetha on the occasion of his 60th birthday 1 Introduction and main result
More informationBank-Laine functions with periodic zero-sequences
Bank-Laine functions with periodic zero-sequences S.M. ElZaidi and J.K. Langley Abstract A Bank-Laine function is an entire function E such that E(z) = 0 implies that E (z) = ±1. Such functions arise as
More informationOn Bank-Laine functions
Computational Methods and Function Theory Volume 00 0000), No. 0, 000 000 XXYYYZZ On Bank-Laine functions Alastair Fletcher Keywords. Bank-Laine functions, zeros. 2000 MSC. 30D35, 34M05. Abstract. In this
More informationProblem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS
Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series
More informationJ. Nonlinear Funct. Anal (2016), Article ID 16 Copyright c 2016 Mathematical Research Press.
J. Nonlinear Funct. Anal. 2016 2016, Article ID 16 Copyright c 2016 Mathematical Research Press. ON THE VALUE DISTRIBUTION THEORY OF DIFFERENTIAL POLYNOMIALS IN THE UNIT DISC BENHARRAT BELAÏDI, MOHAMMED
More informationMeromorphic solutions of Painlevé III difference equations with Borel exceptional values
Zhang Journal of Inequalities and Applications 2014, 2014:330 R E S E A R C H Open Access Meromorphic solutions of Painlevé III difference equations with Borel exceptional values Jilong Zhang * * Correspondence:
More informationEntire functions sharing an entire function of smaller order with their shifts
34 Proc. Japan Acad., 89, Ser. A (2013) [Vol. 89(A), Entire functions sharing an entire function of smaller order with their shifts By Xiao-Min LI, Þ Xiao YANG Þ and Hong-Xun YI Þ (Communicated by Masai
More informationProperties of higher order differential polynomials generated by solutions of complex differential equations in the unit disc
J o u r n a l of Mathematics and Applications JMA No 37, pp 67-84 (2014) Properties of higher order differential polynomials generated by solutions of complex differential equations in the unit disc Zinelâabidine
More informationMEROMORPHIC FUNCTIONS SHARING FOUR VALUES WITH THEIR DIFFERENCE OPERATORS OR SHIFTS
Bull. Korean Math. Soc. 53 206, No. 4, pp. 23 235 http://dx.doi.org/0.434/bkms.b50609 pissn: 05-8634 / eissn: 2234-306 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES WITH THEIR DIFFERENCE OPERATORS OR SHIFTS
More informationOn multiple points of meromorphic functions
On multiple points of meromorphic functions Jim Langley and Dan Shea Abstract We establish lower bounds for the number of zeros of the logarithmic derivative of a meromorphic function of small growth with
More informationDerivation of Some Results on the Generalized Relative Orders of Meromorphic Functions
DOI: 10.1515/awutm-2017-0004 Analele Universităţii de Vest, Timişoara Seria Matematică Inormatică LV, 1, 2017), 51 61 Derivation o Some Results on te Generalized Relative Orders o Meromorpic Functions
More informationNON-ARCHIMEDEAN BANACH SPACE. ( ( x + y
J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. ISSN(Print 6-0657 https://doi.org/0.7468/jksmeb.08.5.3.9 ISSN(Online 87-608 Volume 5, Number 3 (August 08, Pages 9 7 ADDITIVE ρ-functional EQUATIONS
More informationComplex Difference Equations of Malmquist Type
Computational Methods and Function Theory Volume 1 2001), No. 1, 27 39 Complex Difference Equations of Malmquist Type Janne Heittokangas, Risto Korhonen, Ilpo Laine, Jarkko Rieppo, and Kazuya Tohge Abstract.
More informationA Special Type Of Differential Polynomial And Its Comparative Growth Properties
Sanjib Kumar Datta 1, Ritam Biswas 2 1 Department of Mathematics, University of Kalyani, Kalyani, Dist- Nadia, PIN- 741235, West Bengal, India 2 Murshidabad College of Engineering Technology, Banjetia,
More information1 Introduction, Definitions and Notations
Acta Universitatis Apulensis ISSN: 1582-5329 No 34/2013 pp 81-97 THE GROWTH ESTIMATE OF ITERATED ENTIRE FUNCTIONS Ratan Kumar Dutta Abstract In this paper we study growth properties of iterated entire
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationDifference analogue of the lemma on the logarithmic derivative with applications to difference equations
Loughborough University Institutional Repository Difference analogue of the lemma on the logarithmic derivative with applications to difference equations This item was submitted to Loughborough University's
More informationMEROMORPHIC FUNCTIONS THAT SHARE TWO FINITE VALUES WITH THEIR DERIVATIVE
PACIFIC JOURNAL OF MATHEMATICS Vol. 105, No. 2, 1983 MEROMORPHIC FUNCTIONS THAT SHARE TWO FINITE VALUES WITH THEIR DERIVATIVE GARY G. GUNDERSEN It is shown that if a nonconstant meromorphic function /
More informationEstimates for derivatives of holomorphic functions in a hyperbolic domain
J. Math. Anal. Appl. 9 (007) 581 591 www.elsevier.com/locate/jmaa Estimates for derivatives of holomorphic functions in a hyperbolic domain Jian-Lin Li College of Mathematics and Information Science, Shaanxi
More informationDifference analogue of the Lemma on the. Logarithmic Derivative with applications to. difference equations
Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations R.G. Halburd, Department of Mathematical Sciences, Loughborough University Loughborough, Leicestershire,
More informationTheorem Let J and f be as in the previous theorem. Then for any w 0 Int(J), f(z) (z w 0 ) n+1
(w) Second, since lim z w z w z w δ. Thus, i r δ, then z w =r (w) z w = (w), there exist δ, M > 0 such that (w) z w M i dz ML({ z w = r}) = M2πr, which tends to 0 as r 0. This shows that g = 2πi(w), which
More informationON MÜNTZ RATIONAL APPROXIMATION IN MULTIVARIABLES
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXVIII 1995 FASC. 1 O MÜTZ RATIOAL APPROXIMATIO I MULTIVARIABLES BY S. P. Z H O U EDMOTO ALBERTA The present paper shows that or any s sequences o real
More informationAbstract. 1. Introduction
Chin. Ann. o Math. 19B: 4(1998),401-408. THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS** Liu Taishun* Ren Guangbin* Abstract 1 4 The authors obtain the growth and covering
More informationDiscrete Mathematics. On the number of graphs with a given endomorphism monoid
Discrete Mathematics 30 00 376 384 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On the number o graphs with a given endomorphism monoid
More informationGROWTH OF SOLUTIONS TO HIGHER ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS IN ANGULAR DOMAINS
Electronic Journal of Differential Equations, Vol 200(200), No 64, pp 7 ISSN: 072-669 URL: http://ejdemathtxstateedu or http://ejdemathuntedu ftp ejdemathtxstateedu GROWTH OF SOLUTIONS TO HIGHER ORDER
More informationClassification of effective GKM graphs with combinatorial type K 4
Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005,
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl o Inequlities in Pure nd Applied Mthemtics http://jipm.vu.edu.u/ Volume 6, Issue 4, Article 6, 2005 MROMORPHIC UNCTION THAT SHARS ON SMALL UNCTION WITH ITS DRIVATIV QINCAI ZHAN SCHOOL O INORMATION
More informationDeficiency And Relative Deficiency Of E-Valued Meromorphic Functions
Applied Mathematics E-Notes, 323, -8 c ISSN 67-25 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Deficiency And Relative Deficiency Of E-Valued Meromorphic Functions Zhaojun Wu, Zuxing
More informationTelescoping Decomposition Method for Solving First Order Nonlinear Differential Equations
Telescoping Decomposition Method or Solving First Order Nonlinear Dierential Equations 1 Mohammed Al-Reai 2 Maysem Abu-Dalu 3 Ahmed Al-Rawashdeh Abstract The Telescoping Decomposition Method TDM is a new
More informationOn Adjacent Vertex-distinguishing Total Chromatic Number of Generalized Mycielski Graphs. Enqiang Zhu*, Chanjuan Liu and Jin Xu
TAIWANESE JOURNAL OF MATHEMATICS Vol. xx, No. x, pp. 4, xx 20xx DOI: 0.650/tjm/6499 This paper is available online at http://journal.tms.org.tw On Adjacent Vertex-distinguishing Total Chromatic Number
More informationNon-real zeroes of real entire functions and their derivatives
Non-real zeroes of real entire functions and their derivatives Daniel A. Nicks Abstract A real entire function belongs to the Laguerre-Pólya class LP if it is the limit of a sequence of real polynomials
More informationON THE GROWTH OF LOGARITHMIC DIFFERENCES, DIFFERENCE QUOTIENTS AND LOGARITHMIC DERIVATIVES OF MEROMORPHIC FUNCTIONS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 7, July 009, Pages 3767 379 S 000-99470904663-7 Article electronically published on February 0, 009 ON THE GROWTH OF LOGARITHMIC DIFFERENCES,
More informationSome Results on the Growth Properties of Entire Functions Satifying Second Order Linear Differential Equations
Int Journal of Math Analysis, Vol 3, 2009, no 1, 1-14 Some Results on the Growth Properties of Entire Functions Satifying Second Order Linear Differential Equations Sanjib Kumar Datta Department of Mathematics,
More informationJ. M. Almira A MONTEL-TYPE THEOREM FOR MIXED DIFFERENCES
Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 75, 2 (2017), 5 10 J. M. Almira A MONTEL-TYPE THEOREM FOR MIXED DIFFERENCES Abstract. We prove a generalization o classical Montel s theorem or the mixed dierences
More informationMEROMORPHIC FUNCTIONS SHARING ONE VALUE AND UNIQUE RANGE SETS
E. MUES AND M. REINDERS KODAI MATH. J. 18 (1995), 515-522 MEROMORPHIC FUNCTIONS SHARING ONE VALUE AND UNIQUE RANGE SETS E. MUES AND M. REINDERS Abstract We show that there exists a set S with 13 elements
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 381 (2011) 506 512 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Inverse indefinite Sturm Liouville problems
More informationOn the Bank Laine conjecture
On the Bank Laine conjecture Walter Bergweiler and Alexandre Eremenko Abstract We resolve a question of Bank and Laine on the zeros of solutions of w + Aw = 0 where A is an entire function of finite order.
More informationEXPANSIONS IN NON-INTEGER BASES: LOWER, MIDDLE AND TOP ORDERS
EXPANSIONS IN NON-INTEGER BASES: LOWER, MIDDLE AND TOP ORDERS NIKITA SIDOROV To the memory o Bill Parry ABSTRACT. Let q (1, 2); it is known that each x [0, 1/(q 1)] has an expansion o the orm x = n=1 a
More informationOn the Bank Laine conjecture
J. Eur. Math. Soc. 19, 1899 199 c European Mathematical Society 217 DOI 1.4171/JEMS/78 Walter Bergweiler Alexandre Eremenko On the Bank Laine conjecture Received August 11, 214 and in revised form March
More informationStrong Lyapunov Functions for Systems Satisfying the Conditions of La Salle
06 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 004 Strong Lyapunov Functions or Systems Satisying the Conditions o La Salle Frédéric Mazenc and Dragan Ne sić Abstract We present a construction
More informationSOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS
International Journal o Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 2017, 179-187 DOI: 10.28924/2291-8639-15-2017-179 SOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM
More informationResearch Article Subnormal Solutions of Second-Order Nonhomogeneous Linear Differential Equations with Periodic Coefficients
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 416273, 12 pages doi:10.1155/2009/416273 Research Article Subnormal Solutions of Second-Order Nonhomogeneous
More informationFluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs
Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr
More informationFinite Dimensional Hilbert Spaces are Complete for Dagger Compact Closed Categories (Extended Abstract)
Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 www.elsevier.com/locate/entcs Finite Dimensional Hilbert Spaces are Complete or Dagger Compact Closed Categories (Extended bstract)
More informationGROWTH PROPERTIES OF COMPOSITE ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES IN UNIT POLYDISC FROM THE VIEW POINT OF THEIR NEVANLINNA L -ORDERS
Palestine Journal o Mathematics Vol 8209 346 352 Palestine Polytechnic University-PPU 209 GROWTH PROPERTIES OF COMPOSITE ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES IN UNIT POLYDISC FROM THE VIEW POINT
More informationSection 3.4: Concavity and the second Derivative Test. Find any points of inflection of the graph of a function.
Unit 3: Applications o Dierentiation Section 3.4: Concavity and the second Derivative Test Determine intervals on which a unction is concave upward or concave downward. Find any points o inlection o the
More informationCentral Limit Theorems and Proofs
Math/Stat 394, Winter 0 F.W. Scholz Central Limit Theorems and Proos The ollowin ives a sel-contained treatment o the central limit theorem (CLT). It is based on Lindeber s (9) method. To state the CLT
More informationEXISTENCE OF ISOPERIMETRIC SETS WITH DENSITIES CONVERGING FROM BELOW ON R N. 1. Introduction
EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI Abstract. In this paper, we consider the isoperimetric problem in the space R with
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationInteractions between Function Theory and Holomorphic Dynamics
Interactions between Function Theory and Holomorphic Dynamics Alexandre Eremenko July 23, 2018 Dedicated to Walter Bergweiler on the occasion of his 60-th birthday It is not surprising that in the study
More informationFeedback Linearization
Feedback Linearization Peter Al Hokayem and Eduardo Gallestey May 14, 2015 1 Introduction Consider a class o single-input-single-output (SISO) nonlinear systems o the orm ẋ = (x) + g(x)u (1) y = h(x) (2)
More informationBounds of Hausdorff measure of the Sierpinski gasket
J Math Anal Appl 0 007 06 04 wwwelseviercom/locate/jmaa Bounds of Hausdorff measure of the Sierpinski gasket Baoguo Jia School of Mathematics and Scientific Computer, Zhongshan University, Guangzhou 5075,
More informationAdditive functional inequalities in Banach spaces
Lu and Park Journal o Inequalities and Applications 01, 01:94 http://www.journaloinequalitiesandapplications.com/content/01/1/94 R E S E A R C H Open Access Additive unctional inequalities in Banach spaces
More informationarxiv: v3 [math-ph] 19 Oct 2017
ABSECE OF A GROUD STATE FOR BOSOIC COULOMB SYSTEMS WITH CRITICAL CHARGE YUKIMI GOTO arxiv:1605.02949v3 [math-ph] 19 Oct 2017 Abstract. We consider bosonic Coulomb systems with -particles and K static nuclei.
More informationA property of the derivative of an entire function
A property of the derivative of an entire function Walter Bergweiler and Alexandre Eremenko February 12, 2012 Abstract We prove that the derivative of a non-linear entire function is unbounded on the preimage
More informationComptes rendus de l Academie bulgare des Sciences, Tome 59, 4, 2006, p POSITIVE DEFINITE RANDOM MATRICES. Evelina Veleva
Comtes rendus de l Academie bulgare des ciences Tome 59 4 6 353 36 POITIVE DEFINITE RANDOM MATRICE Evelina Veleva Abstract: The aer begins with necessary and suicient conditions or ositive deiniteness
More informationYURI LEVIN AND ADI BEN-ISRAEL
Pp. 1447-1457 in Progress in Analysis, Vol. Heinrich G W Begehr. Robert P Gilbert and Man Wah Wong, Editors, World Scientiic, Singapore, 003, ISBN 981-38-967-9 AN INVERSE-FREE DIRECTIONAL NEWTON METHOD
More informationMATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 9 SOLUTIONS. and g b (z) = eπz/2 1
MATH 85: COMPLEX ANALYSIS FALL 2009/0 PROBLEM SET 9 SOLUTIONS. Consider the functions defined y g a (z) = eiπz/2 e iπz/2 + Show that g a maps the set to D(0, ) while g maps the set and g (z) = eπz/2 e
More informationUNIQUENESS OF NON-ARCHIMEDEAN ENTIRE FUNCTIONS SHARING SETS OF VALUES COUNTING MULTIPLICITY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 4, April 1999, Pages 967 971 S 0002-9939(99)04789-9 UNIQUENESS OF NON-ARCHIMEDEAN ENTIRE FUNCTIONS SHARING SETS OF VALUES COUNTING MULTIPLICITY
More informationSolution of the Synthesis Problem in Hilbert Spaces
Solution o the Synthesis Problem in Hilbert Spaces Valery I. Korobov, Grigory M. Sklyar, Vasily A. Skoryk Kharkov National University 4, sqr. Svoboda 677 Kharkov, Ukraine Szczecin University 5, str. Wielkopolska
More informationPoles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy
Publ. RIMS, Kyoto Univ. 40 2004), 471 485 Poles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy By Shun Shimomura Abstract The first Painlevé hierarchy, which is a sequence of higher
More informationRoot Arrangements of Hyperbolic Polynomial-like Functions
Root Arrangements o Hyperbolic Polynomial-like Functions Vladimir Petrov KOSTOV Université de Nice Laboratoire de Mathématiques Parc Valrose 06108 Nice Cedex France kostov@mathunicer Received: March, 005
More informationOn the Weak Type of Meromorphic Functions
International Mathematical Forum, 4, 2009, no 12, 569-579 On the Weak Type of Meromorphic Functions Sanjib Kumar Datta Department of Mathematics University of North Bengal PIN: 734013, West Bengal, India
More informationGrowth properties at infinity for solutions of modified Laplace equations
Sun et al. Journal of Inequalities and Applications (2015) 2015:256 DOI 10.1186/s13660-015-0777-2 R E S E A R C H Open Access Growth properties at infinity for solutions of modified Laplace equations Jianguo
More informationSpectral radii of graphs with given chromatic number
Applied Mathematics Letters 0 (007 158 16 wwwelseviercom/locate/aml Spectral radii of graphs with given chromatic number Lihua Feng, Qiao Li, Xiao-Dong Zhang Department of Mathematics, Shanghai Jiao Tong
More informationThe Hausdorff measure of a class of Sierpinski carpets
J. Math. Anal. Appl. 305 (005) 11 19 www.elsevier.com/locate/jmaa The Hausdorff measure of a class of Sierpinski carpets Yahan Xiong, Ji Zhou Department of Mathematics, Sichuan Normal University, Chengdu
More information