Value distribution of certain differential polynomials

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PII. S0161171201011036 http://ijmms.hindawi.com © Hindawi Publishing Corp.

VALUE DISTRIBUTION OF CERTAIN

DIFFERENTIAL POLYNOMIALS

INDRAJIT LAHIRI

(Received 5 October 2000 and in revised form 1 April 2001)

Abstract.We prove a result on the value distribution of diﬀerential polynomials which improves some earlier results.

2000 Mathematics Subject Classiﬁcation. 30D35.

1. Introduction and deﬁnitions. Letfbe a transcendental meromorphic function in the open complex plane. The problem of possible Picard values of derivatives of freduces to the problem of whether certain polynomials in a meromorphic function and its derivatives necessarily have zeros. We do not explain the standard deﬁnitions and notations of value distribution theory as those are available in [6].

Deﬁnition_1.1. _{A meromorphic function “}_a_{” is said to be a small function of}_f_if T (r , a)=S(r , f ).

Deﬁnition_1.2_{(see [}₁_, ₄_,₁₀_]). _Let_n₀_{j, n}₁_{j, . . . , nkj} _{be nonnegative integers. The} expressionMj[f ]=(f )n0j_(f(1)₎n1j···_(f(k)₎nkj_{is called a diﬀerential monomial}

gen-erated byfof degreeγM_j=k

i=0nijand weightΓMj= k

i=0(i+1)nij. The sumP [f ]=l

i=1bjMj[f ] is called a diﬀerential polynomial generated byf of degreeγP =max{γM_j : 1≤j ≤l} and weight ΓP =max{ΓMj : 1≤j ≤l}, where T (r , bj)=S(r , f )forj=1,2, . . . , l.

The numbersγ_P =min{γM_j: 1≤j≤l}andk(the highest order of the derivative of finP [f ]) are called, respectively, the lower degree and order ofP [f ].

P [f ]is said to be homogeneous ifγP=γ_P.

AlsoP [F ]is called a quasi diﬀerential polynomial generated byf if, instead of as-sumingT (r , bj)=S(r , f ), we just assume thatm(r , bj)=S(r , f )for the coeﬃcients bj(j=1,2, . . . , l).

Deﬁnition_1.3. _Let_m_{be a positive integer. We denote by}_{N(r , a}_;_f| ≤_m)₍_{N(r , a}_;

f| ≥m)) the counting function of thosea-points off whose multiplicities are not greater (less) thanm, where eacha-point is counted according to its multiplicity.

In a similar manner, we deﬁneN(r , a;f|< m)andN(r , a;f|> m).

Also N(r , a;f| ≤m), N(r , a;f| ≥m), N(r , a;f|< m), and N(r , a;f|> m) are deﬁned similarly, where in counting thea-points offwe ignore the multiplicities.

Finally, we agree to takeN(r , a;f| ≤ ∞)≡N(r , a;f )andN(r , a;f| ≤ ∞)≡N(r , a;f ).

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with proper multiplicities, which are the b-points of g with multiplicities greater thanm.

Deﬁnition_1.5_{(see [}₂_]). _Let_m_{be a positive integer. We denote by}_{Nm(r , a}_;_{f )}_the counting function ofa-points off, where ana-point of multiplicityµ is countedµ times ifµ≤mandmtimes ifµ > m.

As the standard convention, we mean byN(r , f )andN(r , f )the counting functions N(r ,∞;f )andN(r ,∞;f ), respectively.

Hayman [5] proved the following theorems.

Theorem1.6. Iffis a transcendental meromorphic function andn (≥5)is a posi-tive integer, thenψ=f−afn_{assumes all ﬁnite values inﬁnitely often.}

Theorem_1.7. _If_f_{is a transcendental meromorphic function and}_{n (}≥₃₎_{is a}

pos-itive integer, thenψ=ffn _{assumes all ﬁnite values, except possibly zero, inﬁnitely} often.

Whenfis transcendental, entire conclusions of Theorems1.6and1.7hold, respec-tively forn≥3 (cf. [5]) andn≥1 (cf. [3]).

To study the value distribution of diﬀerential polynomials Yang [7] proved the fol-lowing results.

Theorem 1.8. Letf be a transcendental meromorphic function with N(r , f )= S(r , f ), and letψ=fn₊_{P [f ], where}_{n (}_≥₂₎_{is an integer and}_{P [f ]}_{is a} diﬀeren-tial polynomial generated byfwithγP ≤n−2. Thenδ(a;ψ) <1fora≠0,∞.

Theorem _1.9. _Let_f _{be a transcendental meromorphic function with} _{N(r , f )}=

S(r , f ), and letψ=fn_{P [f ], where}_{n (}_≥₂₎_{is an integer and}_{P [f ]}_{is a diﬀerential} polynomial generated byf. Thenδ(a;ψ) <1fora≠0,∞.

Improving all the above results, Yi [9] proved the following theorem.

Theorem1.10. Letfbe a transcendental meromorphic function andQ₁[f ],Q₂[f ] be two diﬀerential polynomials generated byf such thatQ1[f ]≡0,Q2[f ]≡0, and P [f ]=nj=0ajfj (an≡0), wherea1, a2, . . . , an are small functions off. IfF =P [f ] Q1[f ]+Q2[f ], then

n−γQ₂T (r , f )≤N(r ,0;F )+Nr ,0;P [f ]+ΓQ2−γQ2+1

N(r , f )+S(r , f ). (1.1)

InTheorem 1.10we see that the inﬂuence ofQ1[f ]on the value distribution ofF is ignored. In this paper, we show thatTheorem 1.10can further be improved if the inﬂuence ofQ1[f ]is taken into consideration. Throughout, we ignore zeros and poles of any small function offbecause the corresponding counting function is absorbed inS(r , f ).

2. Lemmas. In this section, we present some lemmas which will be needed in the sequel.

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q1∗, q∗2, . . . , qs∗andq1, q2, . . . , qt, respectively. Further letP [f ]=

n

j=0ajfj(an≡0) and γQ≤n. IfP [f ]Q∗[f ]=Q[f ], then

mr , Q∗[f ]≤ s

j=1

mr , q_j∗+ t

j=1

mr , qj+S(r , f ). (2.1)

Lemma2.2. LetQ[f ]=l_j₌₁bjMj[f ]be a diﬀerential polynomial generated byf of order and lower degree k andγ_Q, respectively. Ifz0is a zero offwith multiplicityµ (> k)andz0is not a pole of any of the coeﬃcientsbj(j=1,2, . . . , l), thenz0is a zero ofQ[f ]with multiplicity at least(µ−k)γ_Q.

Proof. Clearlyz₀is a zero ofMj[f ]with multiplicity µn0j+(µ−1)n1j+···+(µ−k)nkj

=µγM_j−ΓMj−γMj

=(µ−k)γM_j+(k+1)γM_j−ΓMj ≥(µ−k)γM_j≥(µ−k)γ_Q.

(2.2)

Sincez0 is assumed not to be a pole of the coeﬃcientsbj (j=1,2, . . . , l)we see thatz0is a zero ofQ[f ]with multiplicity at least(µ−k)γ_Q. This proves the lemma.

Lemma_2.3_{(see [}₁_]). _{The following inequality holds:}

Nr , P [f ]≤γPN(r , f )+ΓP−γP

N(r , f )+S(r , f ). (2.3)

Lemma_2.4_{(see [}₇_]). _Let_{P [f ]}=n_i₌₀_aifi_{, where}_an(≡₀_{), an}₋₁_{, . . . , a}₁_{, a}₀_{are small} functions off. Thenm(r , P [f ])=nm(r , f )+S(r , f ).

Lemma_2.5_{(see [}₄_]). _If_{Q[f ]}_{is a diﬀerential polynomial generated by}_f _with arbi-trary meromorphic coeﬃcientsqj(1≤j≤n), then

mr , Q[f ]≤γQm(r , f )+ n

j=1

mr , qj+S(r , f ). (2.4)

Lemma2.6(see [8]). IfP [f ]is as inLemma 2.4, thenT (r , P [f ])=nT (r , f )+S(r , f ).

3. The main result. In this section, we present the main result of the paper.

Theorem_3.1. _Let_f_{be a transcendental meromorphic function in the open complex} plane, andQ1[f ] (≡0),Q2[f ] (≡0)be two diﬀerential polynomials generated byf such thatkandγ_Q

1be the order and lower degree ofQ1[f ], respectively andP [f ]= n

i=0aifi, wherean(≡0),an−1, . . . , a0are small functions off. If

F=P [f ]Q1[f ]+Q2[f ], (3.1)

then

N(r , f )

−γN(r ,0;f )−Nk+1(r ,0;f )

+S(r , f ), (3.2)

whereγ=γ_Q

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Proof. _If_{n < γQ}

2, the theorem is obvious. So we suppose thatn≥γQ2.

Diﬀeren-tiating (3.1) we get

F=P[f ]Q1[f ]+P [f ]Q 1[f ]+Q

2[f ], (3.3)

whereP[f ]=(d/dz)P [f ]andQi[f ]=(d/dz)Qi[f ]fori=1,2. Multiplying (3.1) by(F/F ), and substituting in (3.3) we get

P [f ]Q∗[f ]=Q[f ], (3.4)

where

Q∗[f ]= _F

F − P[f ]

P [f ]

Q1[f ]−Q

1[f ], (3.5)

Q[f ]=Q2[f ]−

_F

F

Q2[f ]. (3.6)

First we suppose that Q∗[f ]≡0. ByLemma 2.1, it follows from (3.4) thatm(r , Q∗[f ])=S(r , f )becauseγQ=γQ2≤n.

SinceP [f ]=Q[f ]/Q∗[f ], we get byLemma 2.5and the ﬁrst fundamental theorem

mr , P [f ]≤mr , Q[f ]+mr ,0;Q∗[f ]

≤γQ₂m(r , f )+mr , Q∗[f ]+Nr , Q∗[f ]−Nr ,0;Q∗[f ]+S(r , f ) =γQ₂m(r , f )+Nr , Q∗[f ]−Nr ,0;Q∗[f ]+S(r , f ).

(3.7)

So byLemma 2.4

n−γQ₂m(r , f )≤Nr , Q∗[f ]−Nr ,0;Q∗[f ]+S(r , f ). (3.8)

From (3.5) we see that possible poles ofQ∗[f ]occur at the poles off and zeros ofF andP [f ]. Also we note that the zeros ofF andP [f ]are at most simple poles ofQ∗[f ]. Letz0be a pole of f with multiplicityµ. Thenz0is a pole ofQ[f ]with multiplicity not exceeding(µ−1)γQ₂+ΓQ2+1=µγQ2+ΓQ2−γQ2+1 andz0is a pole

ofP [f ]with multiplicitynµ. Hence, from (3.4) it follows thatz0is a pole ofQ∗[f ] with multiplicity not exceedingµγQ2+ΓQ2−γQ2+1−nµ=ΓQ2−γQ2+1−(n−γQ2)µ.

Therefore

Nr , Q∗[f ]≤N(r ,0;F )+Nr ,0;P [f ]+ΓQ2−γQ2+1

N(r , f )

−n−γQ2

N(r , f )+S(r , f ). (3.9)

Now we note that the order of the diﬀerential polynomialQ1[f ]is k+1. Let z0 be a zero off with multiplicityµ > k+1. Letγ_Q

1≥1. Then byLemma 2.2, we see that

z0is a zero ofQ1[f ]with multiplicity at least(µ−1)γ_Q

1. Alsoz0may be a pole of

(F/F )−P[f ]/P [f ]with multiplicity not exceeding 1. Soz0is a zero of((F/F )− P[f ]/P [f ])Q1[f ]with multiplicity at least(µ−k)γ_Q

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Since the lower degree ofQ1[f ]isγQ1, it follows fromLemma 2.2thatz0is a zero

ofQ1[f ]with multiplicity at least(µ−k−1)γ_Q

1.

Thereforez0is a zero ofQ∗[f ]with multiplicity at least(µ−k−1)γ_Q

1. Hence

Nr ,0;Q∗[f ]

≥Nr ,0;Q∗[f ]|f=0, > k+1

≥γ_Q

1N

r ,0;f|> k+1−γ_Q

1(k+1)N(r ,0;f > k+1)+S(r , f ) =γ_Q

1N(r ,0;f )−γQ1

N(r ,0;f| ≤k+1)+(k+1)N(r ,0;f|> k+1)+S(r , f ). (3.10)

So

Nr ,0;Q∗[f ]≥γ_Q

1

N(r ,0;f )−Nk+1(r ,0;f )+S(r , f ). (3.11)

Ifγ_Q

1=0, inequality (3.11) obviously holds. Now from (3.8), (3.9), and (3.11) we get

n−γQ2

T (r , f )≤N(r ,0;F )+Nr ,0;P [f ]+ΓQ2−γQ2+1

N(r , f )

−γ_Q

1

N(r ,0;f )−Nk₊1(r ,0;f )

+S(r , f ). (3.12)

Next we suppose thatQ∗[f ]≡0. Then from (3.4) it follows thatQ[f ]≡0, and so using (3.1) we getP [f ]Q1[f ]=cQ2[f ], wherecis a nonzero constant. Then in a similar line of calculation for inequalities (3.8), (3.9), and (3.11) we get

n−γQ₂m(r , f )≤Nr , Q1[f ]

−Nr ,0;Q1[f ]

+S(r , f ),

Nr , Q1[f ]

≤ΓQ2−γQ2+1

N(r , f )−n−γQ₂N(r , f )+S(r , f ),

Nr ,0;Q1[f ]

≥γ_Q

1

N(r ,0;f )−Nk+1(r ,0;f )

+S(r , f ).

(3.13)

Now from (3.13) we get

N(r , f )

−γ_Q

1

N(r ,0;f )−Nk+1(r ,0;f )

+S(r , f ). (3.14)

This proves the theorem.

Remark_3.2. _{The following example shows that}_{Theorem 3.1}_{is sharp.}

Example _3.3. _Let _f = _ez−_2, _{P [f ]}= _f+_2, _Q₁_{[f ]}= _f_{, and} _Q₂_{[f ]} = _{1. Then} F=P [f ]Q1[f ]+Q2[f ]=(ez−1)2andk=0,γ_Q

1=1,γQ2=0,n=1. Also we see that

n−γQ2

T (r , f )=N(r ,0;F )+Nr ,0;P [f ]+ΓQ2−γQ2+1

N(r , f )

−γ_Q

1

N(r ,0;f )−Nk₊1(r ,0;f )

+S(r , f ). (3.15)

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Theorem_4.1. _Let_f _{be a transcendental meromorphic function and}_Q₁_{[f ] (}≡₀_), Q2[f ] (≡0)be two diﬀerential polynomials generated byf. LetF=fnQ1[f ]+Q2[f ] and

lim sup r→∞

N(r ,0;f )+ΓQ2−γQ2+1

N(r , f )

T (r , f ) < n−γQ2. (4.1)

ThenΘ(a;F ) <1for any small functiona (≡ ∞, Q2[f ])off.

Theorem4.2. LetF=fnQ[f ], whereQ[f ]is a diﬀerential polynomial generated byfandQ[f ]≡0. If

lim sup r→∞

N(r ,0;f )+N(r , f )

T (r , f ) < n, (4.2)

thenΘ(a;F ) <1, wherea (≡0,∞)is a small function off.

Considering the following examples, Yi [9] claimed that Theorems4.1and4.2are sharp.

Example _4.3. _Let _f = _(e4z+₁_)/(e4z−₁₎_, _Q₁_{[f ]} = _1, _Q₂_{[f ]} = _f−_{1, and} _F = f4_Q

1[f ]+Q2[f ]. Thenn=4,γQ2=1,ΓQ2=2, and

lim sup r→∞

N(r ,0;f )+ΓQ2−γQ2+1

N(r , f )

T (r , f ) =n−γQ2. (4.3)

Also we see thatΘ(0;F )=1.

Example4.4. Letf=(ez−1)/(ez+1),Q₁[f ]=1,F=fnQ₁[f ], wheren=2. It is easy to verify that

lim sup r→∞

N(r ,0;f )+N(r , f )

T (r , f ) =n (4.4)

andΘ(1;F )=1.

The following examples suggest that some improvements of Theorems4.1and4.2

are possible.

Example_4.5. _Let_f=_((ez−₁_)/(ez+₁₎₎2_,_Q₁_{[f ]}=_f_,_Q₂_{[f ]}=_{1, and}_F=_{f Q}₁_{[f ]}+ Q2[f ]. Thenn=1,γ_Q

1=1,γQ2=0,ΓQ2=0, and the order of the diﬀerential

polyno-mialQ1[f ]is zero. Clearly

lim sup r→∞

N(r ,0;f )+ΓQ2−γQ2+1

N(r , f )

T (r , f ) =n−γQ2. (4.5)

Also we see thatΘ(1;F )=Θ(∞;F )=3/4,Θ(2;F )=1/2 and so, by Nevanlinna’s three small functions theorem (cf. [6, page 47]),Θ(a;F )≤2−3/4−1/2=3/4 for any small functiona (≡1,2,∞). However, we note that

lim sup r→∞

N(r ,0;f| ≤1)+ΓQ2−γQ2+1

N(r , f )

T (r , f ) =

1

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Example_4.6. _Let_f=_((ez−₁_)/(ez+₁₎₎2_,_{Q[f ]}=_f_{, and}_F=_{f Q[f ]}_{. Then}_n=_1, γ_Q=1, and the order of the diﬀerential polynomialQ[f ]is zero. Clearly

lim sup r→∞

N(r ,0;f )+N(r , f )

T (r , f ) =n (4.7)

andΘ(a;F ) <1 for any small functionaoff. We note that

lim sup r→∞

N(r ,0;f| ≤1)+N(r , f ) T (r , f ) =

1

2< n. (4.8)

The following two theorems improve Theorems4.1and4.2.

Theorem4.7. Letf be a transcendental meromorphic function andQ₁[f ],Q₂[f ] be two diﬀerential polynomials generated byf which are not identically zero. LetF= fn_Q

1[f ]+Q2[f ]. If

lim sup r→∞

Nr ,0;f| ≤χQ1

+ΓQ2−γQ2+1

N(r , f )

T (r , f ) < n−γQ2, (4.9)

thenΘ(a;F ) <1for any small functiona (≡ ∞, Q2[f ])off, where

χQ1=   

1+k ifγ_Q

1≥1, ∞ ifγ_Q

1=0,

(4.10)

andkis the order of the diﬀerential polynomialQ1[f ].

Theorem_4.8. _Let_f_{be a transcendental meromorphic function and}_{Q[f ] (}≡₀₎_be a diﬀerential polynomial generated byf. IfF=fn_{Q[f ]}_and

lim sup r→∞

Nr ,0;f| ≤χQ+N(r , f )

T (r , f ) < n, (4.11)

thenΘ(a;F ) <1for every small functiona (≡0,∞)off, where

χQ=   

1+k ifγ_Q≥1,

∞ ifγ_Q=0, (4.12)

andkis the order of the diﬀerential polynomialQ[f ].

Remark_4.9. _{Theorem 4.7}_{improves Theorems}_1.8_and_4.1_{, and}_{Theorem 4.8}

im-proves Theorems1.9and4.2.

Remark4.10. The following examples show that Theorems4.7and4.8are sharp.

Example4.11. Letf =ez−1,Q₁[f ]=f−f,Q₂[f ]=2f, andF =f2Q₁[f ]+ Q2[f ]. Thenn=2,k=1,ΓQ2=2,γQ2=1, and

lim sup r→∞

N(r ,0;f| ≤2)+ΓQ2−γQ2+1

N(r , f )

T (r , f ) =n−γQ2. (4.13)

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Example_4.12. _Let_f=_ez+_1,_{Q[f ]}=_f−_f_{, and}_F=_{f Q[f ]}_{. Then}_γ

Q=1,k=1, n=1, and

lim sup r→∞

N(r ,0;f| ≤2)+N(r , f )

T (r , f ) =n. (4.14)

Also we see thatΘ(1;F )=1.

As other applications of Theorem 3.1, we obtain the following results which im-prove Theorems1.6and1.7.

Theorem_4.13. _Let_f_{be a transcendental meromorphic function, and}_F=_f−_afn_,

wherea (≡0)is a small function off. Ifn (≥5)is an integer, thenΘ(b;F )≤4/nfor any small functionb (≡ ∞)off.

Theorem4.14. Letf be a transcendental meromorphic function. IfF=fnfand n (≥3)is an integer, thenΘ(a;F )≤4/(n+2)for any small functiona (≡0,∞)off.

We prove Theorems4.8and4.14only.

Proof ofTheorem_4.8. _{First we treat the case} _γ

Q ≥ 1. Then byTheorem 3.1 we get

nT (r , f )≤N(r , a;F )+Nr ,0;P [f ]+N(r , f )

−γ_QN(r ,0;f )−Nk+1(r ,0;f )

+S(r , f ) ≤N(r , a;F )+N(r ,0;f )−N(r ,0;f )

+Nk+1(r ,0;f )+N(r , f )+S(r , f ),

(4.15)

that is,

nT (r , f )≤N(r , a;F )+N(r ,0;f| ≤k+1)+N(r , f )+S(r , f ). (4.16)

Now we treat the caseγ_Q=0. Then fromTheorem 3.1we get

nT (r , f )≤N(r , a;F )+N(r ,0;f )+N(r , f )+S(r , f ). (4.17)

Combining (4.16) and (4.17), we obtain

nT (r , f )≤N(r , a;F )+Nr ,0;f| ≤χQ+N(r , f )+S(r , f ) (4.18)

from which the theorem follows.

Proof ofTheorem_4.14. _{Proceeding in the line of the proof of}_{Theorem 4.8}_we

get

nT (r , f )≤N(r , a;f )+N(r ,0;f| ≤k+1)+N(r , f )+S(r , f ), (4.19)

that is,

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Now by Lemmas2.3and2.5we see that

T (r , F )≤(n+2)T (r , f )+S(r , f ). (4.21)

If possible letΘ(a;F ) >4/(n+2). Then there exits anε (>0)such that for all large values ofr

N(r , a;F ) < _n₋₂

n+2−ε

T (r , F ). (4.22)

From (4.20), (4.21), and (4.22) we get

ε(n+2)T (r , f )≤S(r , f ), (4.23)

which is a contradiction. This proves the theorem.

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Indrajit Lahiri: Department of Mathematics, University of Kalyani, West Bengal 741235, India