022012   Fang M L;Lahiri I (Dep of Applied Mathematics, Dong Hua Univ, Shanghai 200051 P.R. China) : Unique range set for certain meromorphic functions. Indian J Math 2003, 45(2), 141-50.

The uniqueness of meromorphic functions has been studied and the following theorem has been proved. Let S ={z:zⁿ+az^z-1+b=0), where n (≥7) is positve integer and a, b are two constants such that zⁿ + az^n-1 + b=0 has no multiple roots if and f and g b two monoconstant meroconstant meromorphic functions having no simple pole if E (S.f) = E (S.g), E (∞, g) then f=g. This result extends some results of Yi and Lahiri.

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022011   El-Sayied H K (Dep of Math, Fac of Sci, Tanta Univ, Tanta, Egypt) : Complete metric of positive scalar curvature on non compact manifold. Bull Calcutta Mathl Soc 2003, 95(1), 31-8.

A non compact manfold with some modifications to admit a complete metric of positive scalar curvature has been studied. This kind of manifold plays an important role in the existence and non existence of the solution for non compact parabolic equations.

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022010   Duraisamy C;Ramaswamy R S (Mathematics Dep, Kongu Engn Coll, Perundurai-638 052) : On a solution of one step integral equation. Appl Sci Period 2003, 5(2), 78-85.

Approximate numerical solutions of an one-step integral equation are given. This procedure is illustrated through two examples.

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022009   Dong Zhe (Dep of Mathematics, Zhejiang Univ, Hangzhou 310 027) : Rank one operators in reflexive one-sided A-submodules. Proc Indian Acad Sci-Mathl Sci 2004, 114(1), 55-63.

Reflexive one-sided A-submodules U of unital operator algebra A in B (H) have been charaterised completely. Furthermore the invariante subspace lattice Lat ℜ and the reflexive hull Ref ℜ, where ℜ is submodule generated by rank-one operators in U; in particular if L is subspace lattice. The rank-one algebraℜ of Alg L is big enough to determined AlgL in the following senses: Alg L = Alg Lat ℜ and Alg L = Ref ℜ.

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022008   Dheena P;Nandakumar P (Dep of Mathematics, Annamalai Univ, Annamalinagar 608 002 Tamil Nadu) : On regular semirings. Indian J Math 2003, 45(2), 135-40.

1-(2-) regular semirings have been introduced. Equivalent conditions are obtained for a semiring to be 1-(2-) regular. It has been shown tht if R is a nilpotent-free semiring then R is 2-regular if R is strong regular.

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022007   Devendra Kumar (Dep of Math, D.S.M. Coll, Kanth 244 501) : Approximation of generalised axisymmetric potentials having fast growth. Indian J Math 2003, 45(3), 265-78.

Growth and approximation of solution of certain elliptic partial differental equations has been dealt with. These solutions are called Generalized Axisymmetric Potentials (GAPs). The GASPs are taken to be regular in a finite hyperball and the influence of the growth of their maximum moduli on the rate of decay of their approximation errors in supnorm is studied. The characterisations of the q-order and lower q-order of a GASP, interms of the coefficients occuring in its ultra spherical harmonic expansion and rate of decay of their approximation errors have been obtained. The results of Fryant and Gilbert are not applicable, when GASPs are not entire. An attempt has been made to study the growth of such GASPs.

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022006   Deo S D;Thengane K D (Mathematics Dep, N S Sci Coll, Bhadrawati-442 902) : N.dimensional static plane symmetric space.time in bimetric relativity. Appl Sci Period 2003, 5(2), 86-8.

Nil contribution of the perfect fluid distribution to the n-dimensional static plane symmetric space-time in bimetric relativity is obtained. And it is observed that the resulting space time can be reduced to the conformal space-time.

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022005   Datta S;Shukla M (Dep of Math & Astron, Lucknow Univ, Lucknow-226007) : Drag of flow past a cylinder with slip. Bull Calcutta Mathl Soc 2003, 95(1), 63-72.

The solution of the two dimensional flow produced by the translation of a cyllinder with uniform velocity U have been attempted in an incompressible viscous fluid of infinite extent, using the slip boundary condition. The stream function is then used to find the drag on the cylinder and results obtained for a few special cases viz. source, doublet shear flow and hyperbolic flow. It is found that the slippage on the cylinder reduces the drag.

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022004   Das A K (Dep of Mathematics, North-Eastern Hill Univ, Permanent Campus) : Cobordism independence of Grassmann manifolds. Proc Indian Acad Sci-Mathl Sci 2004, 114(1), 33-8.

It has been proved that for F = R C or H Bordism classes of all non bounding Grassmannian manifolds G_k (F^n+k) with k<60> n and having real dimension d, constitute a linearly indepdent set in the unoriented bordism group η_d regarded as a Z₂-vector space.

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022003   Choudhary M P (Mathematics Dep, Univ of Delhi, Delhi-110 007) : Infinite index property of subgroups in a free group. Vikram mathl J 2002, 22, 6-12.

The property of subgroups of infinite index has been studied in a free group. It has been proved that for a finitely generated subgroup of infinite index in a free group Fm, in a definite statistical meaning, the normal subgroup generated by any chosen elements r₁......r_n of Fm has trivial intersection with H.

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022002   Burnwal A P (Mathematics Dep, B I T, Sindri-828 123, Distt-Dhanbad) : Insight into different areas of mathematical modeling. Appl Sci Period 2003, 5(2), 105-7.

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022001   Bhagat R P;Kumar R B (Mathematics Dep, A S Coll, Bikramganj-802 212) : Existence of invariant measure for the flow. Math Educ 2004, 38(1), 47-9.

Characterization of a.p.minimal set and existence of invariant measure on a.p. minimal set in the flow have been given.

022000   Berland M (8, rue Dupin 75006 Paris, , ) : Convergence D'une certaine Serie D' exponentielles. J Analysis 2003, 11, 61-7.

Some properties have been studied on the convergence of a class of exponential series a little more general than those introduced by the author, where in the coefficient are entire functions of bounded index, in the meaning of B. Lepson, with at most, a finite number of zeros. Methods of M.Blambert R. Parvatham and Gerd H. Fricke, which are relatives to the existence of entire functions of bounded Borelien orders and bounded index have been used.

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021999   Bajaj N (Mathematics Dep, Dr Harisingh Gour Vishwavidyalaya, Sagar-470 003) : A fixed point theorem in Banach space. Vikram mathl J 2001, 21, 7-10.

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021998   Babujee J B (Mathematics Dep, Rajiv Gandhi Coll of Engn and Technol, Kirumampakkam, Pondicherry-607 402) : Bimagic labelling in path graphs. Math Educ 2004, 38(1), 12-16.

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021997   Ashok Kumar;Sharan L K;Singh S R (Mathematics Dep, F S College, Fatehpur Sanda, Arwal) : On parametric and strong projective convergence. Appl Sci Period 2003, 5(2), 118-20.

021996   Arora S C;Batra R (Dep of Mathematics, Univ of Delhi, Delhi 110007) : On generalized slant toeplitz operators. Indian J Math 2003, 45(2), 121-34.

For an integer k≥2. a slant Toeplitz operator Uϕ of k^th-order, with symbolϕ in L^∞ (T) is an operator on L² (T) whose representing matrix M = (α_ij) is given by α_ij = [ϕ, z^ki-j], where <60>.,.<62> is the usual inner product in the space L²(T). Some properties like compactness, hyponormality and isometry of k^th-order slant Toeplitz operators and their compressions are discussed. It is also proved that the adjoint of k^th order slant Toeplitz operator is not a k^th -order slant Toeplitz operator.

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021995   Ansari M S A (Town Inter Coll, , Sultanganj, Patna-800 006) : Coincide uniform structures of abelian topological groups. Math Educ 2004, 38(1), 50-1.

Two uniform structures on an (Additive) topological group in general have been discussed. For abelian topological groups, the left and right uniform structures coincide.

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021994   Amardeep Singh;Garg R (Mathematics Dep, Govt Motilal Vigyan Mahavidayalaya, Bhopal-462 003) : Some unique common fixed point theorems in semi-Hausdorff spaces. Vikram mathl J 2001, 21, 42-8.

Some unique fixed point theorems have been established in more generalized semi-Hausdorf spaces, with a weaker condition of continuity. Recent result of Chugh and Rani have been extented and generalized in semi-Haudorff spaces.

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021993   Alimohammadi D;Ebadian A (Dep of Mathematics, Univ of Arak, Arak, 879, I.R. Iran) : Second dual of real Lipschitz algebras of complex functions. J Analysis 2003, 11, 85-103.

A compact matric space (k,d) has been considered where α ∈ (0,1) and where Δ{(x,y)∈ K x K : x=y}, V= (K x K) -Δ, W = K U V. The real Banach algebra of all complex Lipschitz functions (Lip(K, τ,α),<215><215> .<215><215>_/α)_/ and (Lip(K, T,α),<215><215>.<215><215>_/α)_/, where τ is a d-isometry topoological involution on K have been defined. It has been shown that there is a real linear isometry of the real Branch space (lip(K, τ,α),<215><215> .<215><215>_α) into the real Banach space C₀ (W,τ^{<126>,<215><215>< F255><215>.<215><215><215>, where τ<126> is the toplogical involution on W induced by τ. Also, the second dual of (Lip(K, τ,α),<215><215> .<215><215>_α) is isometrically isomorphic to (Lip_R (K,τ,α), <215><215>.<215><215>< v>α), Lip_R (K,τ,α) is the set of all real functions in Lip(K,τ,α).}

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021992   Alice K V;Krishna Kumar V;Padmanabhan A (Dep of Mathematics, Newman College, Thodupuzha 685 585) : On the limit-classifications of even and odd order formally symmetric differential expressions. Proc Indian Acad Sci-Mathl Sci 2004, 114(1), 65-78.

The formally symmetric differential expression M [.] of any order (odd or even) ≥ 2. has been considered. The dimension of the quotient space D(T_max)/D(T_min with M [.] in terms of the behaviour of the determinants det [[ƒ_rg_s](∞)] _rs,εNn where 1≤ n ≤ (order of the expression + 1); here [ƒg](∞) = lim[fg](x)_x→∞, where [ƒg](x) is sesquilinear from in ƒ g assoicated with M has been raised. These results generalise the well known theorem that M is in the limit-point at ∞ if and only if [ƒg](∞)=0 for every ℜ,g ε the maximal domainΔ associated with M.

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021991   Alice K V;Krishna Kumar V (Dep of Mathematics, Newman College, Thodupuzha, Kerala-685585) : Two classes of second and third-order differential expressions in the limit circle case. J Analysis 2003, 11, 115-45.

Two special classes of second and third-order formally symmetric differential expressions whose coefficients are determined by a single function and two real parameters have been considered. Necessary and sufficient conditions on this function and the parameters for the expressions to be in the corresponding limit-circle cases at ∞ have been obtained. It is interesting to note that the limit classifications depend on the relative values of the two parameters.

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000040   Weiyi Z (College of Mathematics, Physics and Information Science, Zhejiang Normal Univ, Jinhua, Zhejiang, P.R. China) : On the general gauss sums and its fourth power mean. Far East J mathl Sci 2003, 10(3), 241-6.

Studies the asymptotic property of the fourth power mean of the general Gauss sums, and give an interesting asymptotic formula.

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000039   Todjihounde L (NO, Inst De Math Et De Sci Phys, B.P. 613 Porto-Novo, Rep. Du Benin) : Ricci deformation and conformal change of metrics. Bull Calcutta Mathl Soc 2003, 95(5), 361-6.

The problem of the existence on a compact Riemannian manifold of a Ricci flow which preserves the conformal class of the initial metric has been examined.

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000038   Tan A (NO, Alabama A & M University, Normal, AL 35762, USA) : Centripetal acceleration in curvilinear motion. Math Educ 2004, 38(2), 63-70.

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000037   Soner N D;Chaluvaraju B (Math Dep, Univ Mysore, Manasagangotri, Mysore-570 006) : The total nonsplit domination number of a graph. Math Educ 2004, 38(2), 77-80.

The total dominating set D of a connected graph G = (V, E) is a total nonsplit dominating set, if the induced sub graph {V-D} is connected. The total nonsplit domination number γ_ts (G) of G is the minimum cardinality of a total nonsplit dominating set of G. Some of the properties of this new parameter are developed.

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000036   Singh N K (University Dep Mathematics, L.N. Mithila University, Darbhanga-846 004) : On generalisation of stochastic integral. Math Educ 2004, 38(2), 109-11.

The properties of stochastic integral as derived by A. V. Skorohad have been discussed. Anticipating integrand has been extended replacing it with smoothness criteria based on chain rule.

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000035   Shenan G M (Math Dep, Fac of Sci, JNV Univ, Jodhpur-342 005) : Application of fractional differintegral operators to certain class of analytic functions with positive coefficients. Bull Calcutta Mathl Soc 2003, 95(5), 377-84.

A class, consisting of functions regular in a unit disk, is introduced such that certain fraction differintegral operator assumes the character of starlike functions of order α. Also sharp coefflicient estimates have been determined for some function in the said class of functions and the maximization is mentioned.

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000034   Sadegheih A (Dep of Industrial Engineering, Univ of Yazd, P.O. Box 89195-741 Iran) : Optimisation using simulated anealing and evolutionary algorithms. Far East J mathl Sci 2003, 10(3), 247-54.

Simulated annealing is an intelligent approach designed to give a good though not necessarily optimal solution, within a reasonable computation time. The motivation for simulated annealing comes from an analogy between the physical annealing of solid materials and optimisation problem. Evolutionary alogrithms are computer-based probelm-solving systems based on principles of evolutionary theroy. A variety of evolutionary alogrithms have been developed and they all share a common conceptual base of simulating the evolution of individual structures via processes of selection, mutation and recombination. The processess depend on the perceived performance of the individual structures as defined by an environment. This paper presents the optimal design of insepctin station in manufacturing systems. The tas is to determine the optimal settings of the production parameters to minimise a cost function. Two artificial intelligence techniques, evolutionary alogrithms and simulated annealing are proposed to search for solutions to these optimisation problems. The perofrmance of each techniques is studied and the results compared with these from conventional methods.

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000033   Rehman A;Basumatary S K;Ahmed M (Dep of Sericulture, Govt of Assam, Goalpara-783101) : Comparison of rhizosphere and non-rhizosphere microflora of som (Machilus bombycina and Soalu (Litsea polyantha) Plants growing in golapara district Assam, India. Envir Ecol 2003, 21(3), 617-9.

Mycological scanning of the rhizophere and non-rhizosphere soil samples collected from som and soalu plantation in the different places of Goalpara district revealed the presence of 13 genera of fungi. Fungal populations varied according to the season, plants variety, soil condition and age of the plants.

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000032   Pravin J M (Electronics and Communication Engineering, Anna University, Chennai-600 025) : Matrix modelling of fibonacci series. Math Educ 2004, 38(2), 106-8.

A Matrix Model for the general term of Fibonacci series is discussed and the general formula for the n^th term of Fibonacci series for second and third order is derived using matrix method. A general procedure using matrices is also given for deriving the general term for an m^th order Fibonacci series.

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000031   Prasad A V M;Sastri V V S (Math Dep, Sree Vidyanikethan Engng Coll, A. Rangampet, Chandragiri Mandal Chittoor district. A.P. 517 102) : An asymptoic formula for partitions into cube full integers. Bull Calcutta Mathl Soc 2003, 95(5), 367-76.

Using a theorem of Richmond and assuming Riemann Hypothesis an asymptotic formula has been obtained for the number of partitions into cube full integers.

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000030   Phaneendra T (Math Dep, Mahatma Gandhi Inst of Technol, Chaithanya Bharathi, Gandipet-500 075) : Generalization of Brian Fisher's theorem. Math Educ 2004, 38(2), 92-4.

If S and T are self-maps of a complete metric space (X, d) satisfying the inequality : d(Sx, TSy) ≤c. max (d(x, Sx), d(x, Sy), D(Sy, TSy), 1/2 [d(x, TSy) + d(Sx, Sy)]} for all x, y εX, where )≤ c <60> 1, Brian Fisher has proved that if either S or T is continuous then S and T have a unique common fixed point. A generalization of this result has been obtained by relaxing the completeness of X and dropping the condition of continuity.

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000029   Pant R P;Jha K;Pande V P (Math Dep, Kumaon Univ, D.S.B. Campus, Nainital-263 002) : Common fixed points by altering distances between points. Bull Calcutta Mathl Soc 2003, 95(5), 421-8.

A common fixed point theorem for four mappings by altering distances between the points under a φ-contractive condition has been obtained.

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000028   Pant L M;Ganju A (HQ Snow and Acalanche Study Establishment, , Manali-175 103) : Fuzzy rule-based system for prediction of direct action avalanches. Curr Sci 2004, 87(1), 99-104.

Rule-based system are widely being used in decision making, control systems and forecasting. In the real world much of the knowledge is imprecise, uncertain, ambiguous and inexact in nature. Fuzzy logic offers a better way to represent complicated situations in terms of simple natural language. An attempt has been made to develop a rule-base for prediction of direct action avalanches of Chowkibal-Tangdhar road axis (Jammu and Kashmir) in Indian Himalaya using fuzzy logic. The condition attributes of the rule-based system are six snow-related parameters selected from the past dataset of the representative observatory 'Stage-II' in the axis. Different fuzzy sets are defined for each parameter on the basis of their distribution with four danger labels of avalanche activity. A total of 101 composite rules are developed for different danger labels of avalanche activity. The results show good agreement with the danger classification for avalanche activity and prediction of non-avalanche activities.

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000027   Majumdar T K;Sengupta A K;Ghosh K (Atapur Kenaram High School, , Atapur, North 24-Parganas) : Torsional vibration of semi-infinite non-homogeneous elastic medium. Math Educ 2004, 38(2), 84-91.

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000026   Kumbhat R K;Daby M S (Math and Statists Dep, J.N.V. Univ, Jodhpur-432 001) : Generating function for biorthogonal polynomial using Lie group theory. Bull Calcutta Mathl Soc 2003, 95(5), 385-90.

Generating functions for biorthogonal polynomial T^{α)_{K-1, n}, (x), defined by Chatterjea (1963), using Weisner (1955) method of Lie group have been obtained.}

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000025   Kong D A;Kim Y H (Dep of Finance and Insurance, Changwon National Univ, Changwon 641773, Korea) : On certain new integral inequalities for continuous functions. Far East J mathl Sci 2003, 10(3), 301-8.

Establishes explicit bounds on certain integral inequalities of the type of Gronwall and its two independent variable generalisation. Some applications are also given to illustrate the usefulness of one of the results.

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000024   Kim B J (Dep of Mathematics, Daejin Univ, Pocheon 487-711, Korea) : An integral equation for the operator valued function space integral. Far East J mathl Sci 2003, 60(3), 309-21.

IN [Nagoya Math. J. 51 (1973), 91-122] and [Proc. London Math. Soc 27 (1973), 345-360], Cameron and Storvick established the existence of the operator valued function sapce integral as an operator from L₁ to L_{∞ for certain functinals. Moreover, they use their function space integral of a functional of the form F(x) = xp <198>⌠^T0 θ(s, x(s))ds<198> to solve an integral equation formally equivalent to Schorodinger's equations. In [J Math. Anal. 50 (1975), 647-667], Johnson and Skoug obtained similar results to those in [Proc. London Math. Soc. 27 (1973), 345-360] with reduced restriction on θ.In [Nagoya Math.J.51(1973),91-122], Cameron and Storvick treated the operator valued functions space integral as an L(L1(R), L∞(R)) thory involving the potential functions of the form F(x) = exp<198>⌠^T0⌠<^ >T0 θ(s, t, x(s), x(t))dsdt)<214>. Obtained an integral equiaton for the operator valued function space integral as a new result for the functinal of the form F(x) = exp<198>⌠^T0⌠<^ >T0 θ(s, t, x(s), x(t))dsdt)<214>, which is formally relatedt o Schrodinger equation.}

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000023   Khobragade N W;Wankhede P C (Math Dep, MJP Educational Campus, Nagpur Univ, Nagpur-440 010) : An inverse unsteady state thermoelastic problem of a thin circular plate in Marchi-Fasulo transform domain. Bull Calcutta Mathl Soc 2003, 95(5), 405-10.

The inverse unsteady-state thermoelastic problem to determine the temperature, displacement and stress functions on the outer curved surface of a thin circular plate of thickness 2 h occupying the space D: O ≤ r ≤ a, -h ≤ = ≤ h is considered by applying finite Marchi-Fasulo transform and Laplace transform techniques.

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000022   Karan S K;Prasad V K (Mathematics Dep, B D Evining College, Patna) : On E(A). semi convex set in a topological vector space. Math Educ 2004, 38(2), 118-19.

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000021   Jha A;Jha K N (University Department of Mathematics, L.N. Mithila University, Darbhanga-846 004) : On Ergodic behaviour of dynamical systems. Math Educ 2004, 38(2), 11214.

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000020   Irmak H;Raina R K;Owa S (Dep of Mathematics, Daejin Univ, Pocheon -487-711, Korea) : Certain results involving inequalities on analytic and univalent functins. Far East J mathl Sci 2003, 60(3), 241-6.

Presents two theorems involving certain inequalities on analytic and univalent functions which are starlike and convex in the unit disc U.

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000019   Irmak H;Owa S (Dep of Mathematics, Akdeniz Univ, 07058-Antalya, Turkey) : Certain inequalities involving analytic and univalent functions. Far East J mathl Sci 2003, 60(3), 353-8.

Presents a theorem involving inequalities on analytic and univalent functions which are close-to-covex in the unit disc. U. Interesting consequencies of our main results are also pointed out.

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000018   Hayashi M (Dep of Mathematics, College of Science and Technology, Nihon Univ, 7-24-1, Narashinonai Funabashi-shi, Chiba -2748501, Japan) : On the local center of generalized lienard-type systems. Far East J mathl Sci 2003, 10(3), 265-83.

Necessary and sufficient conditions for generalized Lienard-type systems x = y, y =- f_{n(x)y^p- (x+gq(x)) to have a local center are given, where fn(x) and gq(x) are real analytic functions.}

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000017   Ganguly A;Jana M (Appl Math Dep, Shri G.S. Inst of Tech & Sci, 23, Park Road, Indore-452 003) : Propagation of spherical MGD shock wave in a non-uniform Protating stellar atmosphere under the action of monochromatic radiation and gravitation. Bull Calcutta Mathl Soc 2003, 95(5), 351-60.

A self-similar, theoretical model of propagation of spherical shock wave in a magneto-gas dynamic rotating non-uniform atmosphere in presence of monochromatic radiation and gravitation is considered. The results discussed depend upon the variations of the flow variables behind the shock which are displayed graphically. A special case is considered in presence and absence of gravitation from which one observes that the influence of gravitation on the variations of radiation flux is significant, where as it does not have a noticeable effect on the other parameters.

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000016   Gaikwad M N;Deshmukh K C (Math Dep, H.R. Arts & Sci Coll, Ashti-442 202, Dist Wardha) : Quasi-static thermal deflection in a thin clamped circular plate. Math Educ 2004, 38(2), 71-6.

The formulation and solution of a quasi-static thermoelastic problem defined as 0≤r≤a, 0≤z≤h, of a thin circular plate has been dealt with when the part of its boundary is kept at zero temperature and the plate initially being subjected to a concentrated heat source located inside the plate. The circular edge is thermally insulated. The expression for temperature and displacement are obtained in a series forms in terms of Bessel's function and they are illustrated numerically.

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000015   Emara S A A (Directorate of Curricula Development, Ministry of Education, P.O. Box 15662, Adliya 336 Manama, Kingdom of Bahrain) : Inqualities in weighted rearragement invariant spaces. Far East J mathl Sci 2003, 10(3), 255-63.

Conditions on pairs of weight functions are obtained for the boundedness of an operator (Kf)(x)=⌠^{∞ k(x,y)f(y)dyon certain rearrangement invariant spaces in terms of conditions on the kernel k(x,y). The results are applied to obtain weighted estimates for the Reimann-Liouville fractional integral operator, the Stirieltjes transform, hte Laplace transform and for some other cases.}

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000014   Dutta T K;Sardar S K (Pure Math Dep, Univ Calcutta, 35, Ballygunge Circular Road, Calcutta-700 019) : On Brown-McCoy semisimple Γ-semirings. Bull Calcutta Mathl Soc 2003, 95(5), 391-404.

The notion of Brown-Mccoy radical or H-radical of a Γ-semiring has been introduced and Brown-McCoy (or H-)semisimple Γ-semirings have been obtained via operator semirings. Also wedderburn-Artin theorem has been obtained for a particular type of Γ-semiring.

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000013   Datta S K (Math Dep, Krishnath Coll, Berhampore, Murshidabad-712 101) : Growth properties of composite entire and meromorphic functions. Bull Calcutta Mathl Soc 2003, 95(5), 429-36.

The growth properties of log T(r,fog) relative to T(r,P<157>f<157>) and log T(R, P<157>f<157>) for sufficiently large R = R (r) where P[f] is a differential monomial generated by a transcendental meromorphic function f.

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