 Advances in Pure Mathematics, 2011, 1, 210-217 doi:10.4236/apm.2011.14037 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Normal Criteria and Shared Values by Differential Polynomials* Jihong Wang1, Qian Lu2, Qilong Liao3 1Department of Mathematics, Southwest University of Science and Technology, Mianyang, China 2Department of Mathematics, Southwest University of Science and Technology, Mianyang, China 3Department of Material Science and Engineer, Southwest University of Science and Technology, Mianyang, China E-mail: wangjihong@swust.edu.cn, luqiankuo1965@ho tmail.com, liaoql@swust.edu.cn Received April 5, 2011; revised April 28, 2011; accepted May 8, 2011 Abstract For a family F of meromorphic functions on a domain D, it is discussed whether F is normal on D if for every pair functionsfz, gzF, nfaf,b and share value on D when , where a, b are two complex numbers, . Finally, the following result is obtained:Let nagg d2, 3n0,aF be a family of meromorphic functions in D, all of whose poles have multiplicity at least 4 , all of whose zeros have multi-plicity at least 2. Suppose that there exist two functions az not idendtically equal to zero, analytic in D, such that for each pair of functions dzf andgin F, 2faz f and 2gazg share the function . If has only a multiple zeros and dzazfz whenever az 0, then F is normal in D. Keywords: Normal Family, Meromorphic Function, Shared Value, Differential Polynomial 1. Introduction and the Main Result In 1959,Hayman proved Theorem 1.1. Letf be meromorphic functions in C, n be a positive integer and a, b be two constant such that , . If 5n0,aandb nfaf b then f is a constant. Corresponding to Theorem 1.1 there is the following theorems which confirmed a Hayman’s well-known con- jecture about normal families in . Theorem 1.2. Let F be a meromorphic function family in D, be a positive integer and a, b be two constant such that . If and for each functionn0,aandb 3nfFn, faf b, then F is normal in D. This result is due to S. Y. Li (), X. J. Li  (), X. C. Pang (), H. H. Chen and M. L. Fang (). 5n5n4n3nIn 2001, M. L. Fang and W. J. Yuan  obtained Theorem 1.3. Let F be a meromorphic function family in D, a, b be two constants such that . If, for each function 0,aandb fF,2fafband the poles of fz are of multiplicity 3 at least, then F is nor-mal in D. Let D be a domain in C, fz be meromorphic on D,and. aC 1:fEaf aDZDfz a Two functions f and g are said to share the value a if fEa Ega. For a case in Theorem 1.2, Q. C. Zhang  improved Theorem 1.2 by the idea of shared values and obtained the following result. 4nTheorem 1.4. Let F be a family of meromorphic func-tions in D, n be a positive integer and a, b be two con-stant such that ,4n0,aandbn. If, for each pair of functions f and g in F, fafand ngag share the value b, then F is normal in D. In this paper, we shall discuss a condition on which F still is normal in D for the case and obtain the following result. 2n3Theorem 1.5. Let F be a family of meromorphic func-tions in D, all of whose poles have multiplicity 2 at least, and a, b be two constant such that . 0,aandb *Supported by China Industrial Technology Development Program (B3120110001). J. H. WANG ET AL. 211If, for each pair of functions f and g in F, 3fafand 3gag share the value b in D, then F is normal in D. We denote #21fzfzfz for the spherical de- rivatives of fz.The following example imply that the restriction of poles in Theorem 1.5 is necessary. Example 1.  Let :1Dzz and nFf, where 1,,1,2,(1)nfz zDnnz n Then for each pair m, n, 3mmff and 3nnffz share the value 0 in D. But F is not normal at 0 since #n1fn. But we also have the following examples which imply that on the same as restriction of poles in Theorem 1.5 F is not normal in D if for each pair of functions fand g in F, 2faf and 2gag share the value b on D. Example 2.  Let 21nfznz znfor 1, 2,n, and :1zz  Clearly, 4210nnfz fnzn , and nfzonly a double pole and a simple zero. Since #0nfn, as from Marty’s criterion we nhave that nfz is not normal in In fact, in the-present paper we also obtain two results as follows. Theorem 1.6. Let F be a family of meromorphic func-tions in D, all of whose poles have multiplicity 4 at least, all of whose zeros have multiplicity 2 at least, and a, b be two constant such that . If, for each pair of functions f and g in F, 0,aandb 2faf and 2gag share the value b in D, then F is normal in D. Theorem 1.7. Let F be a family of meromorphic func-tions in D, all of whose poles have multiplicity at least 4 , all of whose zeros have multiplicity at least 2. Suppose that there exist two functions not idendtically equal to zero, analytic in D, such that for each pair of functions f and g in F, azdz2faz f and 2gaz0azg share the function in D. If az has only a multiple zeros and whenever dzfz then F is normal in D. The following example shows that the condition fzwhen in Theorem 1.7 is necessary. 0azExample 3.  Let :1Dzz and nFf where41,,1,2,nfz zDnnz. We take 34az z and 0dz. Clearly, F fails to be nor-mal at 0z However, all poles of nfzare of multi-plicity 4, and for each pair m, n, 2mmfaz fand 2nnfaz f share analytic functions in dz. 2. Lemmas To prove the above theorems, we need some lemma as follows: Lemma 2.1. ([1,2]) Let fzbe a meromorphic function in C, n be a positive integer and b be a non-zero constant. If nffb, then f is a constant. Moreover if f is a transcendental meromorphic function, then nffz assumes every finite non-zero value finitely often. Lemma 2.2. () Letfz be a transcendental me-romorphic function with finite order in C. If fz has only multiple zeros, then it’s first derivative fassumes every finite value except possibly zero infinitely often. Lemma 2.3. () Letfz be a non-polynomials rational function in C. If fz has only zeros of multi- plicity 2 at least, then 2cz dfaz b where a, b, c, d are four constants, 0, 0ac. Lemma 2.4. () If fzbe a transcendental mero-morphic function in C, then either fz assumes every finite value infinitely often or every derivative()lf as-sumes every finite value except possibly zero infinitely often. If fz is a non-constant rational function and fza, a is a finite value, then ()lf assumes every finite value except possibly zero at least once. Lemma 2.5. () Letfzk be a transcendental meromorphic function with finite order, all of whose ze-roes are of multiplicity at least , and let 1Pz be a polynomial, Pz is not idendtically equal to zero. Then ()kfzPz has infinitely many zeros often. Lemma 2.6. () Letfz be a non-polynomial ra-tional functions in C, all of whose zeroes are of multi-plicity at least 4. Then rfzzhas a zeros at least often. Lemma 2.7. () Let F be a family of meromorphic functions on the unit disc , all of whose zeroes have multiplicity p at least, all of whose poles have multiplic-ity q at least. Let  be a real number satisfying pq. Then F is not normal at a point 0z if and only if there exist 1) points nz, ; 0nzz2) functions nfF; and 3) positive numbers 0n such that Copyright © 2011 SciRes. APM 212 J. H. WANG ET AL.  nn nnnfzg g   spherically uniformly on each compact subset of C, where g is a non-constant meromorphic function satisfying the zeros of g are of multiplicities p at least and the poles of gare of multiplicities q at least. Moreover, the order of g is not greater than 2. 3. Proofs of Theorem 1.5.-1.7. 3.1. Proof of Theorem 1.5. Suppose that there exists one point 0 such that F is not normal at point 0. Without loss of generality we assume that . By Lemma 2.7, there exist points, ,0n, functions zDz00zznz znfFand positive numbers 0nsuch that 11njjjjjgfzg (3.1) spherically uniformly on each compact subset of C, where g is a non-constant meromorphic function with order , all of whose poles are of multiplicities k at least. 2From (3.1) we have  11nnnjjjj jjjnnnnjjjfzaf zbgagb gag    (3.2) By the same method as , from Lemma 2.1 it is not difficult to find that ngag has just a unique zero 0. Set 1g again, if then 3n2nngaga n   thus 2nan  has just a unique zero 0. Thus 0 is a multiple pole of  or else a zero of . 2naIf 0 is a multiple pole of , since 2nna  has only one zero 0, then . By Lemma 2.1 again, 20na is a constant which contradicts with g is not any constant. So we have that  has no multiple poles and a have only a unique zero. By Lemma 2.1, and Lemma 2.4, we have  is not transcendental. If  is non-constant polynomial, then 20lnaA  . Since all zeros of are of multiplicity 2, then . 3lDenoting  for 1n1n,11nn, we have 0lAaand10lAl . Since all zeros of are of multiplicity, then 214n00,. If 00, then 00which contradicts with 0a0 . So  is a constant. Next we prove that there exists no rational functions such as . Noting that 11nn and  has no multiple pole, we may set  121mmmsnnt 1, 112112,()s1nA  (3.3) where A is a non-zero constant,st,12,,,smm m2,,)js are s positive integers,j,(1 . For a convenience of stating, we denote 21 ,,mn12smm mm  (3.4) then 21mns . From (3.3), we have 1111mnA1smsnth 11,pq  (3.5) where 1hmtn12ststa20st a  111m 1smsph 1 ,nta 11nq0  (3.6) are three polynomials. Since has only a unique zero  then there exists a non-zero constant B such that  012lnnt,nB a   (3.7) so 10211 nnntBp 112,l   (3.8) where plnt1ttb21t0b   is a poly-nomial. From (3.5) we also have 1mA2131smsntp211n    (3.9) where 3p is a polynomial also. We denote degp for the degree of a polynomial p, from (3.5) and (3.6) we may obtain degdeghstpm11deg qn11,t t (3.10) Copyright © 2011 SciRes. APM J. H. WANG ET AL. 213tFrom (3.8), (3.9) and (3.10) we may obtain 2deg ,p (3.11) 3deg2 22.pts (3.12) Since has only a unique zero a0 and 21(1,2,,),jmjs then 0j (1 rom (3.8), (3.9) and (3.11) it follows,2,,j.)s F that then , (3.13) Since , then w3deg 1pl22degmsp t  21jmn221mns, so by (3.13) e have st. If lnmt, fro (3.8), (3.9) and (3.12), we have Then, . Combining with above inequality 311deg 222ntlpts 21ts2st, wout a contradiction. nt, then from (3.5) and (3.7) we bring abIf e have that is . If , then 1this is impossible. Thus, l deg degpq 11degmsh nt 2st . So 1mtndegh1 degdeg22mtnsnth tnst hstst   1mtnre, and deg 1hst.Therefo11. Thenm2sttne have m21s, this contradicts to 2. Again from (3.8) and (3.9), wtst. This comple ptes theroof of Theorem 1.5. .2. Proof of Theorem 1.6. or any points , Without loss of generality, we 3 F0zDose thatset 00z. Supp F is not normal at 00z, then by L 2.7, we have that there exist a uence nemma subseqfF, points sequence 0zD, and a positive num-bers n, 0n, such that 1,znnnnngf g (3.14) spherically uniformly on each compact subset of C, where g is a non-constant meromorphic function with or, all of whose poles are ofmultiplicities at least 2, all hose zeros are of multiplicities at least 4. From (3.14) we have der 2of w 222nnn1gagadgg  (3.15) If , then0ga 0gac, this contra-dito which allfcts zeros o g have multiplicity at least 4. If for any point C, 0a, then By Lemma 2.2, we have thatg g is not transcendental in C, sogis non-constantal function in C. By Lemmwe also have that  rationa 2.3  3dabcga contradictions. Therefore, 2gag have a zeros. We may claim that2gag h as a uni- que zero 0. Otherwis*e, suppose that 00, are two distinguis of h zeros2gag sitive nuthen there exists a pomber0such that *00,,NN.On the other han by Hur-n find two point sequences d,witzrem we c’s Theoa0,nN, **0,nNSuch that **00,nn, and nn220nn nggad  *mm2*2 0mmmggad  then, we have2n naf z0nnnnn d nfz, *2m maf z*0mmmmm d mfzev. Fr ypothesis thery pair functions om the hat for f, g in F, 2fz af and 2gzag share comp mber dave lexnu in D, we hz af2n nz0nnnnn dmf, *2m naf z*0n nnnnfz d m. Fix , the0m, let n n 200mmfafd. Since 2mmfzafzd n w0, mhas no accumulation points, so for sufficiently large have *mmzz e 0nnn then *,nnnnnnzz Thiradicts to s cont*00,,NN. Thus, 2gag has zero0a unique. Further- either 0move thatre, we ha is a me poles of ultiplg or 0 is a unzero of ique ga. If 0 is a le poles of gmultip, then 0ga, Cfor any . By Lemma 2 Lemmmediately deduce that .2 anda 2.3, we im-g must be a constant in C, Copyright © 2011 SciRes. APM 214 J. H. WANG ET AL. which contradicts to g is a non-constant mero-morphic functions in C. erefore, g Th has only a simple poles and ga has a un0ique . But since g has otiple poles, so we that nly a mule havg is entire in C and ga has a unique 0. Also by Lemma 2.2, hat we have tg is a nstant polynomials, all of whose zeros mul-tiplicity at least 4. Setting 1non-coare of122sm , 2smmgA ms1211mmwe have g 1A h12   12ssa02sa,AWhere ,hm20, 01,,saa t,s are a1, 2, are some complex constan s, jmjs positive integers, 4jm, and 1jsjj. m mThus, we have gaBl0 , gawhere 3l. So we have that 0Bl1lg00g. If 0g, then 00g0ctions. . Bu 0t 0efore, Fga is norm, a contraadiTher l at 0z. .3. or a3 Proof of ny heTheorem az.7 1.7. 0, by the FzDore, ifplete we may give the comproof of Tm 1 same argument as Theorem 1.6, we emit the detail. In the sequel, we shall prove that F is normal at which 0az. Set raz zbz, where bz is analytic 0, 01b positive integer, 2r. at z, r is a11:,rFFFtion zzffz F zuncFor every fFz in 1F, from the hypothesis in Theorem 1.7, we ca that all zeros of n seeFz are of order at least 4, all poles of Fz are of micity at least 2. Supposeultipl that 1F exis not normal at zn0, then by Lemma 2.7, there ists a subsequence 1FF, a point sequence ,1nnzz r,and a positive nequence number s, n 1z0, such thnnatnnrnn11nzfnnnzngFg on comact subsets ofrmlyprically uni (3.16) C, wsphe fohere g is a non-constant meromorphic function on C, whose zeros are of multiplicity at least 4, and all of whose poles are multiple. Moreover, gall of  has an or-der at most 2. Now we distinguish two cases: Case 1. nnz . Without loss of a generalization, w exe assume that thereists a point zsuch that ,1nzzzr, we have  11()11rrnnnrg2222nnnnnn nnn nnnnrnnn nfzgzgzggzrgznote 1S   (3.17) For the sake of convenience, we defor of the set all zeros of g,2S for the set of all zeros of g, and 3S for t sof all poles ofgheet . Since22nngglim ngg ,  11limnggn, and  uni- formly on compact subsets of 1\CSlim 0nnnrzly on coms of C, thus uniformpact subsetlim nn nfzn, uniformlympact sub- on cosets of 123\CS SS. Thus, it is not difficult to see that  2221nnn nnnnnnnnnnnnnnnzazfzazfzdzdzazfzdgbznnnnnnzf     (3.18) uniformly on compact subsets of. If \CS 123S S10gbz , then gbz , for any 123\CS SS . Thus,  gbz for any C. By Lemma 2.5, we can see that g is not ndental in C, but is a rational functioo from Lemma 2.3, we deduce thatgtransce n. Als is constant, which contradicts to the fact that g is non-constant. On the other hand, it is easy to see that g is not iden-tically equal to bz. Hence, gbe as thez has one zeros at least in Crguments in Theorem 1.5 and Theorem 1.6, we deduce that . In fact, by the sam ag has a unique zero 0. By Lemma 2.5, we ca that n seeg is not trdental in C, so ganscen is non- consttional function in C. For a non-cnt poly-ant raonstaCopyright © 2011 SciRes. APM J. H. WANG ET AL. 215nomials g, and noting that g has only a zero with multiplicity at least 4, we have 0l,3gbz Bl  Thus, 10lgBl . Hence, g has a zero 0 at most. If 0 is a zero of g, then  g0000gg . But 00gbz , In the sequel, a contradiction. we denote r the degree of a pop fodeglynomial p. If g i s non polynomials ra-tional functions, then we set 11m 2121212,stmsnnnt   (3.19) Wher t. t (3.20) Then, m 2jn, stjkqngAe jm4,1,2,,js; 1, 2,,j114, 2jkmms  1111A1111stmmsnnthg11pq  (3.21) where 2 120st ststq a ,deg hshmpAa 1t 112 12 111smmmsh   111tntq 12112nn Since gbz has a uno ique zer0, so we set 101l nt. Then 11() tnntBz  (3.22) where Bnonzero constafrom (3.22), we gb is a have 1102ntBp1ttb221tln (3.23) where 1bis a poly- hat g2deg 20tplqt nomial,pt. follow tFrom (3.21), it 11 2mA113221npg2stmsnt3222 223021st st23stpmqmqcc    is also a polynomial, 3deg22 2pst. cases to derivative a cWe distinguish fiveontradiction: Subcase 1.1. mq. Then from (3.21), we have lqt. So, 222deg, 1pti it, 00deg1, 11hst hhst  and 333deg22 2,122 2pstitst  From (3.23) and (3.24), we have. S also fro 231iie om (3.23) and (3.24), we also hav3deg . Thus, we have 322122lsti st1ilp2 Since lq. tt and2qt, then we have 22si. On the ot, from and (3.24), we her hand (3.23) also have22degms p . Since 4ms, we have 22sti. . Subcase 1.2. mThis is impossible1q. Then lqt, 22deg pti, 2it1, deg 1hst and 333deg222, 1222psti tst Similarly to Subcase (1.1), from (3.23) and (3.24), we also have that 231ii. Also from (3(331deglp , .23) and .24), we have then, we have 221ts i On the ot-larly to the argucase (1.1), from (3.23) and (3.24), we also have her hand, simiment of Sub222degmsp ti, then 221sti . This also is3. 2mq impossible. Subcase 1.. Then we still have 222 1lp tst3 ,deg,1,degqttii th and , 3deg22 2pst. Therefor, 222lst, so2s 2t. Similarly, we 22sti, t22have 2mshen sti. . This is a contradictione 1.4. 1mqSubcas. Thenlqt, deg 1hst, 32, deg22tps and 22deg pt2,i i0t.2.ti From (3.23) and (3.24), we have 2ms Thus,22stiand 221 .ts i This iSubcase 1.5. ms impossible. . Then , 2qlqt  (3.24) where deg 1hst,3deg 2pst22d2gpt, ande. From (3.23) and (3.24), we have 31deg22 1lpst and 22degp t. ms21tsSo, we have that 2 and st. This is a con-tradiction. Case 2. Suppose that there exists a complex number C and a subsequence of sequence 1nnz, still noting it 1znn, such that 1znn. Wea con- have Copyright © 2011 SciRes. APM 216 J. H. WANG ET AL. verges  11ˆnnnnnnnnnnF FzzggH   (3.25) spherically uniform on compact subsets of C. Clearly, all zeros of ˆg are of multiplicity at least 4, all poles of ˆg arultiplicity at least 2. For each 00e of m, it to see that there exists a neighborhood ),is easy(0N of 0, such that  ˆrrnHg, the con bei spherically,vergenceng uniform on 0N. For 00, since 0 is the pole of g, then there exists 0, such tt ha ˆ1g is an on alytic2:D2, 1Hn aytic on re anal:2D2 for sufficiently la- rge n. Since 1Hrnnnnf  then 00 is a zero of ˆ1g has order at least r, we cace that n dedu1rnH converges uniform to lyˆ1rg on 2:2D Hence, we have  11ˆrrnnrnnnGHfg (3.26) spherically uniform on compact subsets of C. It follows that 00G from f whenever 0a for Dall of ze, hence ros of G have orst 4, oles of der at leaall of pG have o least 2. Noting that rder at22rrGb Gd 222nnnnnrnnn nnnnrfaf dGG    (3.27) If , then  20rGGrG0, so rG, 101rGCr  for any C. Since 00G, then. Also 00Csince G ha0, this the zeroiplicity at least 4, then Gs is a contradiction. Therefore,  2rGGs of mult is not identically equal to zero. If  2rGG for any C, then G has no multiple poles and. Note that 0GrG les, so Ghas only multiple po at Gi on Ce ths entire. Also by Lemma 2.5, we hav is not tran-scendental in C, and then G isynomial. Thus, a pol0rGC, where 00C. We have 1rGr  , then from 00G and a multiplicities of every zeros of G it follows that 0G for any C, this Hence, is impossible. 2GGr has some zeros. In fact, by the sagument as the C may dethat me arase 1, weduce 2rGG ha s a unique zero 0. Thus, we ha0ve that either  is multi-ple poles of G or 0 is a unique zero of Gr. ilarlySim, if 0 is multiple poles of G, from that 2rGG has a unique zero 0 it atfollows th rG for any C. By Lemma 2.5, we have that G is not transcendentain by al. AgLemmGa 2.6, we have that  is a constant, is whicha contradiction. Hence, G has nle pole and o multiprG has a e zero 0uniqu. Thus, G is entire on C and rG has a unique zero 0. By Lemma 2.5, we have that G must be a polyno-mial. Setting  22, 11smmsgA  ) where, 12,,,m (3.28smmm are s posntegers, 4m itive ij1, 2,,js , 1sjjmm G0lrB l is a positiv) , .29 (3) ve where e integer, lBl3, we ha1l1lrGr0, (3.30) 1 2(3 101lrGrrBl  . (3.31) For 00G, we have 00 and 0j. From (3.29) it follows that 0j,d (3.31)1, 2,,js. or 1,Fromha (3.29), (3.30)a, fn2,js,, we ve l0rjjBjjrBl1Bll  1jj (3.32) (3.33)011lr r20l2rr, we have  (3.34) From (3.32) and (3.33),jrl 0,1r j2,,s (3.35) If lr, then 00, this is im lrpossible. Therefore, we have, and so Copyright © 2011 SciRes. APM J. H. WANG ET AL. Copyright © 2011 SciRes. APM 217 H. H. Chen and M. L. Fang, “On the Value Distribution of nff,” Science 12 srrl0 in China Series A, Vol. 38, No. 7, c Functions,” Indian Journal of Ma- of Mathematics, 1995, pp. 789-798.  M. L. Fang and W. J. Yuan, “On the Normality for Fami-lies of MeromorphiFrom (3.33) and (3.34), we also have , 121srrl0th thematics, Vol. 43, 2001, pp. 341-350.  W. K. Hayman, “Picard Values of Meromorphic Func-tions and the Its Derivatives,” Annalsen 001rr . Thus, we have 00, a contra-dictFinally, we prove that F is normal at the origin. For any function sequenceion. Vol. 70, 1959, pp. 9-42. doi:10.2307/1969890  W. K. Hayman, “Meromorphic Functions,” Clarendon, Oxford, 1964. nfz in F, since 1F is nor-at , theositivmal n there exist a pe number 0z X. J. Huang and Y. X. Gu, “Normal Families of Mero-morphic Functce knF of nF suc12 and subsequenh that ions with Multiple Zeros and Poles,” Journal of Mathematical Analysis and Applications, Vol. 295, No. 2, 2004, pp. 611-619. doi:10.1016/j.jmaa.2004.03.041  X. J. Huang and Y. X. Gu, “Normmorphic Functions,” Results inknF converges uniformly to a meromorphic function hz or  on 0, 2N. Noting 0nF, we de- duce thate exists a postive number 0M such  therithatal Families of Mero- Mathematics, Vol. 49, knFzM for any 0,zN. Again noting 2006, pp. 279-288. doi:10.1007/s00025-006-0224-2  S. Y. Li, “On Normal Criterion of Meromorphic Func-tions,” Journal of Fujian Normal University, Volthat ve t for 0knf we hahat knfz all 0,zN, that is, knfz is analytic in 0, . 25 hina Series A, Vol. 28, 1985, pp. ence in China Series A, Vol. 33, No. 5, 1990, orphic Functions Omitting a Function ii,” tions with Multiple Ze-erican Mathematical Society, ” Journal of Mathe-N. 1984, pp. 156-158.  X. J. Li, “Proof of Hayman’s Conjecture on Normal Fam-ilies,” Science in CThere kn, we havfore, for all e 112,2kknrrnz zMzFrfz 596-603.  X. C. Pang, “On Normal Criterion of Meromorphic Func-tions,” Sciknfz pp. 521-527.  X. C. Pang, D. G. Yang and L. Zalcman, “Normal Fami-lies of MeromzBy Montel’s Theorem, is normal at , 0and thus F is normal at . The complete proof ofTheorem 1.7 is given. 4. ulo the referee for a numbelpful suggestions to improve the paper. Singularities of Finite Order,” ática Iberoamericana, Vol. 11, N373. 0z Computational Methods and Function Theory, Vol. 2, No.1, 2002, pp. 257-265.  Y. F. Wang and M. L. Fang, “Picard Values and Normal Families of Meromorphic Func Acknowledgements The authors are gratef ter of ros,” Acta Mathematica Sinica, Chinese Series, Vol. 41, No. 4, 1998, pp. 743-748.  L. Zalcman, “Normal Families: New Perspectives,” Bul-letin (New Series) of the Amh . References 5  W. Bergweiler and A. Eremenko, “On thethe Inverse to a Meromorphic Function ofVol. 35, No. 3, 1998, pp. 215-230.  Q. C. Zhang, “Normal Families of Meromorphic Func-tions Concerning Sharing Values, matical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 545-551. doi:10.1016/j.jmaa.2007.05.032 Revista Matem1995, pp. 355-o. 2,