Recent zbMATH articles in MSC 30D35https://www.zbmath.org/atom/cc/30D352022-05-16T20:40:13.078697ZWerkzeugUniqueness of \(L\) function with special class of meromorphic function under restricted sharing of setshttps://www.zbmath.org/1483.111902022-05-16T20:40:13.078697Z"Banerjee, Abhijit"https://www.zbmath.org/authors/?q=ai:banerjee.abhijit"Kundu, Arpita"https://www.zbmath.org/authors/?q=ai:kundu.arpitaSummary: The purpose of the paper is to rectify a series of errors occurred in [\textit{A. Banerjee} and \textit{A. Kundu}, Lith. Math. J. 61, No. 2, 161--179 (2021; Zbl 1469.11341); \textit{P. Sahoo} and \textit{A. Sarkar}, An. Științ. Univ. Al. I. Cuza Iași, Ser. Nouă, Mat. 66, No. 1, 81--92 (2020; Zbl 1474.11153); \textit{Q.-Q. Yuan} et al., Lith. Math. J. 58, No. 2, 249--262 (2018; Zbl 1439.11231)] for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing, which is stronger than the usual one. We manipulate the newly introduced notion in this specific section of literature to resolve all the complications. Not only that we have subtly used the same sharing form to a well known unique range set [\textit{G. Frank} and \textit{M. Reinders}, Complex Variables, Theory Appl. 37, No. 1--4, 185--193 (1998; Zbl 1054.30519)] to settle a long time unsolved problem.Value distribution of \(L\)-functions and a question of Chung-Chun Yanghttps://www.zbmath.org/1483.111922022-05-16T20:40:13.078697Z"Li, Xiao-Min"https://www.zbmath.org/authors/?q=ai:li.xiaomin"Yuan, Qian-Qian"https://www.zbmath.org/authors/?q=ai:yuan.qianqian"Yi, Hong-Xun"https://www.zbmath.org/authors/?q=ai:yi.hongxunSummary: We study the value distribution theory of \(L\)-functions and completely resolve a question from [\textit{L. Yang}, Value distribution theory. Berlin: Springer-Verlag; Beijing: Science Press (1993; Zbl 0790.30018)]. This question is related to \(L\)-functions sharing three finite values with meromorphic functions. The main result in this paper extends corresponding results from [\textit{B. Q. Li}, Proc. Am. Math. Soc. 138, No. 6, 2071--2077 (2010; Zbl 1195.30041)].Value distributions of \(L\)-functions concerning polynomial sharinghttps://www.zbmath.org/1483.111932022-05-16T20:40:13.078697Z"Mandal, Nintu"https://www.zbmath.org/authors/?q=ai:mandal.nintuSummary: We mainly study the value distributions of L-functions in the extended Selberg class. Concerning weighted sharing, we prove an uniqueness theorem when certain differential monomial of a meromorphic function share a polynomial with certain differential monomial of an L-function which improve and generalize some recent results due to \textit{F. Liu} et al. [Proc. Japan Acad., Ser. A 93, No. 5, 41--46 (2017; Zbl 1417.11144)], \textit{W.-J. Hao} and \textit{J.-F. Chen} [Open Math. 16, 1291--1299 (2018; Zbl 1451.11098)] and the author and \textit{N. K. Datta} [``Uniqueness of L-function and its certain differential monomial concerning small functions'', J. Math. Comput. Sci. 10, No. 5, 2155--2163 (2020; \url{doi:10.28919/jmcs/4836})].Uniqueness of differential \(q\)-shift difference polynomials of entire functionshttps://www.zbmath.org/1483.300622022-05-16T20:40:13.078697Z"Mathai, Madhura M."https://www.zbmath.org/authors/?q=ai:mathai.madhura-m"Manjalapur, Vinayak V."https://www.zbmath.org/authors/?q=ai:manjalapur.vinayak-vSummary: In this paper, we prove the uniqueness theorems of differential \(q\)-shift difference polynomials of transcendental entire functions.On meromorphic solutions of nonlinear delay-differential equationshttps://www.zbmath.org/1483.300632022-05-16T20:40:13.078697Z"Mao, Zhiqiang"https://www.zbmath.org/authors/?q=ai:mao.zhiqiang"Liu, Huifang"https://www.zbmath.org/authors/?q=ai:liu.huifangSummary: Using Cartan's second main theorem and Nevanlinna's theorem concerning a group of meromorphic functions, we obtain the growth and zero distribution of meromorphic solutions of the nonlinear delay-differential equation \(f^n(z) + P(z) f^{( k )}(z + \eta) = H_0(z) + H_1(z) e^{\omega_1 z^q} + \cdots + H_m(z) e^{\omega_m z^q}\), where \(n, k, q, m\) are positive integers, \( \eta, \omega_1, \cdots, \omega_m\) are complex numbers with \(\omega_1 \cdots \omega_m \neq 0\), and \(P, H_0, H_1, \cdots, H_m\) are entire functions of order less than \(q\) with \(P H_1 \cdots H_m \not\equiv 0\). Especially for \(\eta = 0\), some sufficient conditions are given to guarantee the above equation has no meromorphic solutions of few poles.Paired Hayman conjecture and uniqueness of complex delay-differential polynomialshttps://www.zbmath.org/1483.300642022-05-16T20:40:13.078697Z"Gao, Yingchun"https://www.zbmath.org/authors/?q=ai:gao.yingchun"Liu, Kai"https://www.zbmath.org/authors/?q=ai:liu.kai.4|liu.kai.1|liu.kai.2|liu.kai|liu.kai.3|liu.kai.5Summary: In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of \(f(z)^nL(g)-a(z)\) and \(g(z)^nL(f)-a(z)\), where \(L(h)\) takes the derivatives \(h^{(k)}(z)\) or the shift \(h(z+c)\) or the difference \(h(z+c)-h(z)\) or the delay-differential \(h^{(k)}(z+c)\), where \(k\) is a positive integer, \(c\) is a non-zero constant and \(a(z)\) is a non-zero small function with respect to \(f(z)\) and \(g(z)\). The related uniqueness problems of complex delay-differential polynomials are also considered.Entire solutions of differential-difference equations of Fermat typehttps://www.zbmath.org/1483.300652022-05-16T20:40:13.078697Z"Hu, Peichu"https://www.zbmath.org/authors/?q=ai:hu.peichu"Wang, Wenbo"https://www.zbmath.org/authors/?q=ai:wang.wenbo"Wu, Linlin"https://www.zbmath.org/authors/?q=ai:wu.linlinSummary: In this paper, we extend some previous works by Liu et al. on the existence of transcendental entire solutions of differential-difference equations of Fermat type. In addition, we also present a precise description of the associated entire solutions.Some results on uniqueness of meromorphic functions concerning differential polynomialshttps://www.zbmath.org/1483.300662022-05-16T20:40:13.078697Z"Husna, V."https://www.zbmath.org/authors/?q=ai:husna.vSummary: In this paper, we study the uniqueness problem of certain differential polynomials generated by two meromorphic functions. The results of the paper extend some recent results due to \textit{C. Meng} and \textit{X. Li} [J. Anal. 28, No. 3, 879--894 (2020; Zbl 1455.30028)].One-way sharing of sets with derivatives and normal functionshttps://www.zbmath.org/1483.300672022-05-16T20:40:13.078697Z"Singh, Virender"https://www.zbmath.org/authors/?q=ai:singh.virender-pal"Lal, Banarsi"https://www.zbmath.org/authors/?q=ai:lal.banarsiSummary: The aim of this paper is to study normal functions under the weaker condition of one-way sharing of sets and further to improve and generalize the earlier work of \textit{Q. Chen} and \textit{D. Tong} [Bol. Soc. Mat. Mex., III. Ser. 25, No. 3, 589--596 (2019; Zbl 1430.30016)], \textit{Y. Xu} and \textit{H. Qiu} [Filomat 30, No. 2, 287--292 (2016; Zbl 1474.30243)].On the extended class of SUPM and their generating URSM over non-Archimedean fieldhttps://www.zbmath.org/1483.300882022-05-16T20:40:13.078697Z"Banerjee, Abhijit"https://www.zbmath.org/authors/?q=ai:banerjee.abhijit"Maity, Sayantan"https://www.zbmath.org/authors/?q=ai:maity.sayantanSummary: In this article, we investigate an extended class of strong uniqueness polynomial over non-Archimedean field than that was recently studied by \textit{H.H. Khoai} and \textit{V. H. An} [\(p\)-Adic Numbers Ultrametric Anal. Appl. 12, No. 4, 276--284 (2020; Zbl 1456.30082)]. We also find the unique range set of weight 2 corresponding to the SUPM which improve and generalize significantly the results of the paper [loc. cit.] and an earlier one due to \textit{P.-C. Hu} and \textit{C.-C. Yang} [Acta Math. Vietnam. 24, No. 1, 95--108 (1999; Zbl 0986.30025)].Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaceshttps://www.zbmath.org/1483.320082022-05-16T20:40:13.078697Z"Quang, Si Duc"https://www.zbmath.org/authors/?q=ai:si-duc-quang.Summary: The purpose of this paper is twofold. The first purpose is to establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into an arbitrary projective variety intersecting a family of hypersurfaces in subgeneral position. In our result, the truncation level of the counting functions is estimated explicitly. Our result is an extension of the classical second main theorem of H. Cartan and is also a generalization of the recent second main theorem of M. Ru and improves some recent results. The second purpose of this paper is to give another proof for the second main theorem for the special case where the projective variety is a projective space, by which the truncation level of the counting functions is estimated better than that of the general case.Growth of solutions of non-homogeneous linear differential equations and its applicationshttps://www.zbmath.org/1483.341202022-05-16T20:40:13.078697Z"Pramanik, Dilip Chandra"https://www.zbmath.org/authors/?q=ai:pramanik.dilip-chandra"Biswas, Manab"https://www.zbmath.org/authors/?q=ai:biswas.manabLet \(H\subset \mathbb{C}\) be set with positive upper density, let \(a_0(z),\ldots,a_k(z)\), \(b(z)\) and \(c(z)\) be entire functions and let \(0\leq q < p\). In the paper under review it is shown that if there exists a constant \(\eta >0\) such that \[ |a_j(z)|\leq e^{q|z|^\eta}, \qquad |b(z)| \geq e^{p|z|^\eta}, \qquad |c(z)| \leq e^{q|z|^\eta}, \] for all \(z\in H\), then all meromorphic solutions \(f\not\equiv 0\) of \[ a_k(z) f^{(k)} + \cdots + a_1(z)f' + a_0(z) f = b(z) f + c(z) \] are of infinite order. The paper is concluded by two results on sharing value problems related to the equation above.
Reviewer: Risto Korhonen (Joensuu)