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The critical points of entire fractions and B.Shapiro's

12th conjecture of entire functions

Lande Ma ( dzy200408@126.com )

Tongji University

Zhaokun Ma ( dzy200408@sina.cn )

Shandong University

Research Article

Keywords:

DOI: https://doi.org/

License: This work is licensed under a Creative Commons Attribution 4.0 International License.

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Additional Declarations: No competing interests reported.

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The critical points of entire fractions and

B.Shapiro’s 12th conjecture of entire functions

Lande Ma1,2* and Zhaokun Ma1,2

1*School of Mathematical Sciences, Tongji University, Siping

Road, Shanghai, 200092, Shanghai, China.

2Yanzhou College, Shandong Radio and TV University, Honghua

Xi Street, Yanzhou, 272100, Shandong, China.

*Corresponding author(s). E-mail(s): dzy200408@126.com;

Contributing authors: dzy200408@sina.cn;

Abstract

For any real polynomial p(x)of even degree k. B. Shapiro propose

the 12th conjecture saying that the sum of the number of real zeros of

two polynomials ((k−1)(p

′(x))2

−kp(x)p

′′ (x)and p(x)) is larger

than 0. We comprehensively prove the original B.Shapiro’s 12th con-

jecture and B.Shapiro’s 12th conjecture of entire functions by showing

all cases that the conjecture are true, and the case that it is not true.

Keywords: entire fractions, critical points, polynomials, zeros

Mathematics Subject Classiﬁcation: Primary 30C15 Secondary 30D20 , 26C10

1 Introduction

It is well known that the theory of entire functions began as a research ﬁeld

from the pioneer work of Laguerre[1], soon after the Weierstrass product rep-

resentation became available. In 1900, in the survey of Borel[2]. It is stated

that one of the core problems of the study of entire functions is ﬁnding rela-

tions between the zeros of a real entire function and the zeros of its derivatives.

Here, an entire function is said to be real if it takes real values on the real axis.

Given a real polynomial p(x), if p(x) has all real and simple zeros, then the

function p(x) is (locally) strictly monotone was known to Gauss[3]. We can

1

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2The critical points of entire fractions and B.Shapiro’s 12th conjecture of entire functions

reformulate it in the form of Laguerre inequality: Let p(x) be a real polynomial

with only real zeroes. Then p(x)p′′ (x)−[p′(x)]2≤0, x∈R. The class of entire

functions that are uniform limits on compact sets of sequences of polynomi-

als with only real zeroes is the Laguerre-P´olya class (LP)[4]. And the entire

functions of the Laguerre-P´olya class satisfy Laguerre inequality.

A reﬁnement of the Laguerre inequality constitutes the Wiman conjec-

ture and P´olya conjectures. The Wiman’s conjecture was eventually conﬁrmed

by Bergweiler, Eremenko and Langley[6]. One P´olya’s conjecture was settled

by Craven, Csordas and Smith[9]. Another P´olya’s conjecture was settled by

Bergweiler, Eremenko[7].

By building upon results that resolved a conjecture of Wiman-P´olya’s type

conjectures, a number of conditions are established under which a real entire

functions fmust belong to the class LP. These conditions typically involve

the diﬀerential polynomial in the form of f(x)f′′ (x)−κ(f′(x))2, where κis a

real number[5]. The zeroes of the function f(x)f′′ (x)−κ(f′(x))2when fis a

meromorphic function have been studied in [8].

Hawaiian conjecture was proposed when studying the P´olya’s conjecture[9],

saying that if p(x) is a real polynomial, then the number of real zeroes of

(p

′(x)

p(x))′does not exceed the number of non-real zeroes of p(x). And the con-

jecture can be extended to entire functions[10]. The Hawaiian conjecture was

proved in 2011 by Tyaglov[11]. Then, B. Shapiro propose three conjectures

around the Hawaiian Conjecture(see Conjectures 11, 12 and 13 in [12]).

The 12th Conjecture by B. Shapiro which states: For any real polynomial

p(x) of even degree. ♯r[(k−1)(p′(x))2−kp(x)p′′ (x)] + ♯rp(x)>0. Here, k

donates the degree of p(x), ♯rp(x) stands for the number of real zeros of a

polynomial p(x) with real coeﬃcients.

In this paper, we extend the 12th Conjecture by B. Shapiro to entire func-

tions as: For any real entire function f(x). ♯r[f(x)f′′(x)−κ(f′(x))2]+♯rf(x)>

0. Here, κis a real number. Especially, when κ=k−1

k,kis the degree of

polynomial f(x), the B.Shapiro’s 12th conjecture of entire functions is 12th

conjecture of B.Shapiro.

Our results show that, in most cases, the conjecture of entire functions is

true.

Theorem1 For any real entire function f(x), when f(x) has real zeros, or

f(x) has no real zeros and has at least one real critical point.

Then the quantity ♯r[f(x)f′′(x)−κ(f′(x))2] +♯rf(x)>0 if and only if one

of the following four cases holds: (1). f(x) has real zeros; (2) f(x) has at least

two real critical points or one real critical point which its exponents is larger

than 1; (3). F F (+∞) = 0, or F F (−∞) = 0. (4). f(x) has one simple real

critical point w, that is f′(x) = C(x)(x−w). Where C(x) is a function with

C(w)= 0, and the function f(x)C′(x)(x−w) + C(x)f(x)−κ(C(x))2(x−w)2

must has real zeros.

Theorem2 For any real entire function f(x), when f(x) has no real zeros

and no real critical points.

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Then the quantity ♯r[f(x)f′′(x)−κ(f′(x))2] + ♯rf(x)>0 if and only if

one of the following two cases holds: (1). FF (+∞) = F F (−∞) = A1. A1

can be zero, constant and also can be the positive or negative inﬁnity. (2).

When F F (x) has real critical points, F F (+∞) = 0, F F (−∞) = A2. Or,

F F (+∞) = A2, F F (−∞) = 0, A2 can be a non-zero constant and also can

be the positive or negative inﬁnity.

For the rest two cases, B. Shapiro’s 12th Conjecture of entire functions are

false, they are Theorem 1.13 and Theorem 1.22. We also give examples to show

cases described in Theorem 1, 2, 1.13 and 1.22 does occur.

2 The results

According to the classical theory of entire functions, the entire functions has

three representations. If f(s)= 0, everywhere, then f(s) = eP(s).P(s) is an

entire functions. If there are ﬁnitely many zeros of f. Then f(s) = (s−z1)(s−

z2)···(s−zk)eP(s). If there are inﬁnitely many zeros of f. The zeros z1,z2,·· ·

of entire functions f(s) of order ρhave the property. P∞

k=1 1

(|zk|)ρ+w<+∞, for

all w > 0. Let pbe the least integer (p≤ρ) such that P∞

k=1 |zk|−p−1<+∞.

The following product representation holds (Hadamard’s theorem on entire

functions).

f(s) = sλeP(s)Q∞

k=1(1 −s

zk) exp ( s

zk+· ·· +sp

pzp

k

).

Where P(s) is a polynomial of degree not exceeding ρ.λ= 0 if f(0) = 0,

and λis the multiplicity of the zero s= 0 if f(0) = 0.

Let G(s) = sλeP(s),Gzl(s) = exp ( s

zk+· ·· +sp

pzp

k

). f(s) = G(s)Qn

l=1(1 −

s

zl)Gzl(s). G(s), Gzl(s) have no zeros and poles. According to the representa-

tion of G(s) and Gzl(s), we can obtain: G′(s) = λsλ−1eP(s)+sλP′(s)eP(s),

G′

zl(s) = ( 1

zl+s

z2

l

+· ·· +sp−1

zp

l

) exp ( s

zl+· ·· +sp

pzp

l

). G′(s) and G′

zl(s) have no

poles.

Lemma 2.1. f′(s)has no poles.

Proof For entire functions which have no any zeros and the ﬁnite number of zeros,

their proofs which p

′

has no poles are all simple, here, we needn’t to give. Observe

that f

′(s)

f(s)=G

′(s)

G(s)+Pn

l=1(G

′

zl(s)

Gzl(s)+(−1

zl)

1−s

zl

) = G

′(s)

G(s)+Pn

l=1(G

′

zl(s)

Gzl(s)+1

s−zl).

f

′

(s) = f(s)( G

′(s)

G(s)+Pn

l=1(G

′

zl(s)

Gzl(s)+1

s−zl)) = f(s)G

′(s)

G(s)+Pn

l=1(G

′

zl(s)

Gzl(s)f(s) +

f(s)1

s−zl) = G

′

(s)Qn

l=1(1 −s

zl)Gzl(s) + Pn

l=1(G

′

zl(s)

Gzl(s)G(s)Qn

j=1(1 −

s

zj)Gzj(s) + 1

s−zlG(s)Qn

j=1(1 −s

zj)Gzj(s)) = G

′

(s)Qn

l=1(1 −s

zl)Gzl(s) +

Pn

l=1(G

′

zl(s)G(s)Qn

j=1,j=l(1 −s

zj)Gzj(s) + 1

s−zlG(s)Qn

j=1(1 −s

zj)Gzj(s)).

G(s) and Gzl(s) have no zeros and poles. G

′

(s) and G

′

zl(s) have no poles. So,

G

′

(s)Qn

l=1(1 −s

zl)Gzl(s) has no poles. G

′

zl(s)G(s)Qn

j=1,j=l(1 −s

zj)Gzj(s) have no

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poles. The factor s−zlin denominator is oﬀset by the factor in Qn

j=1(1 −s

zj), so,

1

s−zlG(s)Qn

j=1(1 −s

zj)Gzj(s) have no poles. So, f

′

(s) has no poles. □

Lemma 2.2. For F F (s) = (f

′(s))β

f(s). Let κ=1

β, the function ∆1 = f′′ (s)f(s)−

κ(f′(s))2. When κ>1, all critical points of F F (x)are all zeros of ∆1. When

κ= 1, all critical points of F F (s)are all zeros of ∆1. When κ<1, all zeros

of f′(s)are all critical points of F F (s). Except zeros of f′(s), all critical points

of F F (s)are all zeros of function ∆1.

Proof F F

′

(s) = ( (f

′(s))β

f(x))

′

=β(f

′(s))β−1f

′′ (s)f(s)−(f

′(s))βf

′(s)

(f(s))2

=β(f

′(s))β−1(f

′′ (s)f(s)−1

β(f

′(s))2)

(f(s))2.

F F

′

(s) = ( (f

′(s))β

f(s))

′

=(f

′(s)) 1

κ

−1(f

′′ (s)f(s)−κ(f

′(s))2)

κ(f(s))2. When κ>1, zeros of

f

′

(s) are all poles of F F

′

(s). All poles of f

′

(s) are all critical points of F F (s).

According to Lemma 1.1, f

′

(s) has no poles, so all critical points of F F (s) are all

zeros of function ∆1.

When κ= 1, poles and zeros of f

′

(s) cannot be zeros and poles of F F

′

(s). All

critical points of F F (s) are all zeros of ∆1.

When κ<1, zeros of f

′

(s) are all critical points of F F (s). So, except zeros of

f

′

(s), all critical points of F F (s) are all zeros of ∆1. □

Lemma 2.3. If f(s)has no real zeros. For f′(s), if its exponent of the real

critical point wof f(s)is larger than 1, then wis a real zero of ∆1.

Proof According to the condition of this lemma, let f

′

(s) = C(s)(s−w)l,wis a real

critical point of f(s). And C(w)= 0, and it is a ﬁnite value. f

′′

(s) = C

′

(s)(s−w)l+

lC(s)(s−w)l−1.

∆1 = f(s)C

′

(s)(s−w)l+lC(s)f(s)(s−w)l−1−κ(C(s))2(s−w)2l

= (s−w)l−1(f(s)C

′

(s)(s−w) + lC(s)f(s)) −κ(C(s))2(s−w)2l.

l > 1, f(w)C

′

(w)(w−w) + lC(w)f(w) = lC(w)f(w)= 0, and lC (w)f(w) is a

ﬁnite value. so, wis still a real zero of ∆1. □

Lemma 2.4. If f(s)has no real zeros and one simple real critical point w,

that is f′(s) = C(s)(s−w), where C(s)is a function with C(w)= 0, then w

is not a zero of ∆1.

Proof The wis critical point of f(s). C(w)= 0, and it is a ﬁnite value. f

′′

(s) =

C

′

(s)(s−w) + C(s).

f(s)f

′′

(s)−κ(f

′

(s))2=f(x)C

′

(s)(s−w) + C(s)f(s)−κ(C(s))2(s−w)2.

f(w)C

′

(w)(w−w) + C(w)f(w)−κ(C(w))2(w−w)2=C(w)f(w)= 0. □

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We derive Theorem 1 and Theorem 1.13 from a series of following lemmas.

Lemma 2.5. When f(x)has real zeros, where xis a real variable, let ∆2 =

f′′ (x)f(x)−κ(f′(x))2.♯r∆2 + ♯rf(x)>0.

When f(x) has no real zeros, ♯rf(x) = 0.

Lemma 2.6. For f(x)which has no real zeros, when f′(x)has at least 2

distinct real zeros, ♯r∆2 + ♯rf(x)>0.

Proof The fraction F F (x) is a real function. f(x) has no real zeros, according to

Lemma 1.1, f

′

(x) has no real poles. For F F (x), there are no real poles, and it satisﬁes

all conditions of Rolle’s theorem, we can use Rolle’s theorem.

According to the condition which f

′

(x) has at least 2 real zeros, so, between two

adjacent real zeros of F F (x), F F (x) has at least one real critical point.

At least one real critical points of F F (x) cannot be the zeros of f

′

(x), according

to Lemma 1.2, at least one real critical point of F F (x) is real zeros of function ∆2.

So, ♯r∆2 ≥1>0. □

We give example 1.7 to show that case described in Lemma 1.6 does occur.

Example 2.7. f1(x) = ex3−3x,κcan be any real number.

f1has no real zeros. f′

1= 3(x2−1)ex3−3x.f′

1has two distinct real zeros. In

the interval (−1,1), there is one real zero of f1f′′

1−κ(f′

1)2.♯r[f1f′′

1−κ(f′

1)2]≥

1>0. And ♯rf1= 0.

Applying Lemma 1.3, obviously, we get Lemma 1.8.

Lemma 2.8. When f(x)has no real zeros and one real critical point. For

f′(x), if the exponent of the critical point of f(x)is larger than 1, then ♯r∆2+

♯rf(x)>0.

We give example 1.9 to show that case described in Lemma 1.8 does occur.

Example 2.9. f2(x) = ex3,κcan be any real number.

f2has no real zeros, f′

2= 3x2ex3.f′

2has one repeated zeros w= 0. For f′

2,

the exponent of the critical point w= 0 is 2 and larger than 1, so, ♯r[f2f′′

2−

κ(f′

2)2]≥1>0. And ♯rf2= 0.

Lemma 2.10. For f(x)has no real zeros and one real simple critical point of

f(x).F F (+∞) = 0, or F F (−∞) = 0.♯r∆2 + ♯rf(x)>0.

Proof The fraction F F (x) satisﬁes all conditions of Rolle’s theorem. f

′

(x) has at

least 2 real zeros, one is a ﬁnite zero of F F (x), another is an inﬁnite zero of F F (x)

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on the real axis. So, between the ﬁnite zero and inﬁnite of F F (x) on the real axis,

F F (x) has at least one real critical point. So, ♯r∆2 ≥1>0. □

We give example 1.11 to show that case described in Lemma 1.10 does

occur.

Example 2.11. f3(x) = ex2,κ= 2.

f′

3= 2xex2.f′

3has no real zeros and has one critical point w= 0, which

exponent is 1. f′′

3= 4x2ex2+ 2ex2.f3f′′

3−κ(f′

3)2= (2 −4x2)ex2has two

real zeros. F F3=√2xe x2

2

ex2=√2xe−x2

2.F F3(+∞) = F F3(−∞) = 0. So,

♯r[f3f′′

3−κ(f′

3)2]≥2>0. And ♯rf3= 0.

Lemma 2.12. For f(x)has no real zeros and one real simple critical point,

that is f′(x) = C(x)(x−w), where C(x)is a function with C(w)= 0.

F F (+∞)= 0 and F F (−∞)= 0. The function f(x)C′(x)(x−w)+C(x)f(x)−

κ(C(x))2(x−w)2has real zeros, ♯r∆2 + ♯rf(x)>0.

Proof In the proof of Lemma 1.4, we have: f(x)f

′′

(x)−κ(f

′

(x))2=f(x)C

′

(x)(x−

w)+C(x)f(x)−κ(C(x))2(x−w)2. When f(x)C

′

(x)(x−w)+C(x)f(x)−κ(C(x))2(x−

w)2has real zeros, ♯r[f(x)f

′′

(x)−κ(f

′

(x))2]>0. So, ♯r∆2 + ♯rf(x)>0. □

For the case which f(x) has real zeros, or f(x) has no real zeros and has at

least one real critical point, except ﬁve cases of Lemma 1.5, Lemma 1.6, Lemma

1.8, Lemma 1.10 and Lemma 1.12, only one case is remained. According to the

result and proof of Lemma 1.12, under this case, B.Shapiro’s 12th conjecture

of entire functions is not true. So, we have:

Theorem 2.13. For any real entire function f(x), when f(x)has no real

zeros and has at least one real simple critical point.

Then the quantity ♯r[f(x)f′′ (x)−κ(f′(x))2] + ♯rf(x) = 0 if and only if

F F (+∞)= 0 and F F (−∞)= 0;f′(x) = C(x)(x−w), where C(x)is a

function with C(w)= 0, and the function f(x)C′(x)(x−w) + C(x)f(x)−

κ(C(x))2(x−w)2has no real zeros.

We give example 1.14 to show that case described in Theorem 1.13 does

occur.

Example 2.14. f4(x) = f3(x) = ex2,κ= 1.

f4f′′

4−κ(f′

4)2= 2e2x2has no real zeros. F F4=2xex2

ex2= 2x.F F4(+∞) =

+∞,F F4(−∞) = −∞. So, ♯r[f4f′′

4−κ(f′

4)2] = 0, and ♯rf4= 0. It is contrast

to Shapiro’s conjecture.

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Applying Lemma 1.5, Lemma 1.6, Lemma 1.8, Lemma 1.10, Lemma 1.12

and Theorem 1.13, we get Theorem 1.

When p(x) is a polynomial of even degree, p(x) has real zeros, or p(x) has

no real zeros and has at least one real critical point. The p(x) satisﬁes the

condition of Theorem 1.13 and Theorem 1, let κ=k−1

k,kstands for degree

of polynomial p(x), the 12th conjecture of B.Shapiro of entire functions is

the original 12th conjecture of B.Shapiro. So, the original 12th conjecture of

B.Shapiro is also proved.

When the entire functions f(x) has no real zeros and no real critical points.

If F F (+∞) and F F (−∞) obtain the positive and negative values with the

diﬀerent positive and negative symbol, the positive and negative values can

contain the positive or negative inﬁnity. According to that FF (x) is continuous

on the real axis, on the real axis, there is at least one zero of F F (x). In the

front, we have already given all results which FF (x) has at least one real zero.

So, this case belongs to the situation which we have already given in front.

We give an example that f(x) has no real zeros and no real critical points.

f(x) = (x2+x+ 4)ex.f′(x) = (x2+ 3x+ 5)ex. Obviously, neither f(x) nor

f′(x) has real zeros. So, in the following, we still need to give the results which

f(x) has no real zeros and no real critical points.

We derive Theorem 2 and Theorem 1.22 from a series of following lemmas.

Lemma 2.15. For f(x)has no real zeros and no real critical points, if

F F (+∞) = F F (−∞) = 0, then ♯r∆2 + ♯rf(x)>0.

Proof Now suppose F F (+∞) = F F (−∞) = 0, according to Rolle’s Theorem, on

the real axis, there is at least one critical point. At least one critical point is the zero

of function ∆2. ♯r∆2 ≥1>0. ♯rf(x) = 0. □

We give example 1.16 to show that case described in Lemma 1.15 does

occur.

Example 2.16. f5(x) = e−ex,κ<1.

f′

5=−e−ex+x.f5has no real zeros and critical points.

F F5=(f

′

5)1

κ

f5= (−1) 1

κe(1−1

κ)ex+x

κ. When (1 −1

κ)<0, κ<1, when

x= +∞,F F5(+∞) = 0. x=−∞,F F5(−∞) = 0. ♯r[f5f′′

5−κ(f′

5)2]+♯rf5>0.

Lemma 2.17. For f(x)has no real zeros and no real critical points, if

F F (+∞) = F F (−∞) = ∞,∞is the inﬁnity which has the same positive and

negative symbol, then ♯r∆2 + ♯rf(x)>0.

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Proof (1

F F (x))

′

=−F F

′(x)

F F 2(x). So, F F (x) and 1

F F (x)have the same critical points.

1

F F (+∞)=1

F F (+∞)= 0. Repeat the proof of Lemma 1.15, we can obtain 1

F F (x)has

at least one real critical point, so, F F (x) has at least one real critical point. □

We give example 1.18 to show that case described in Lemma 1.17 does

occur.

Example 2.18. f6(x) = ex3+3x,κ= 1.

f′

6= (3x2+ 3)ex3+3x.f6has no real zeros and critical points. F F6=f

′

6

f6=

3x2+ 1. F F6(+∞) = F F6(−∞) = +∞.

Lemma 2.19. For f(x)has no real zeros and no real critical points, if

F F (+∞) = F F (−∞) = A,Ais the constant which has the same positive and

negative symbol, then ♯r∆2 + ♯rf(x)>0.

Proof Applying Rolle’s Theorem, F F (x) has at least one real critical point. □

We give example 1.20 to show that case described in Lemma 1.19 does

occur.

Example 2.20. f7(x) = (x2+ 20)ex,κ= 1.

f′

7= (x2+ 2x+ 10)ex.f7has no real zeros and critical points. F F7=f

′

7

f7=

x2+2x+20

x2+20 .F F7(+∞) = F F7(−∞) = +1.

Lemma 2.21. For f(x)has no real zeros and no real critical points, if

F F (+∞) = 0,F F (−∞) = A. Conversely, If F F (+∞) = A,F F (−∞) = 0,A

can be a non-zero constant and it also can be the positive or negative inﬁnity.

When F F (x)has the real critical points, ♯r∆2 + ♯rf(x)>0.

The result of Lemma 1.21 is obvious, we needn’t to give its proof.

For the case which f(x) has real zeros and no real critical points, except

four cases of Lemma 1.15, Lemma 1.17, Lemma 1.19 and Lemma 1.21, only

one case is remained, under this case, 12th conjecture of B.Shapiro of entire

functions is not true. So, we have:

Theorem 2.22. For any real entire function f(x), when f(x)has no real

zeros and no real critical points.

Then the quantity ♯r[f(x)f′′ (x)−κ(f′(x))2] + ♯rf(x) = 0 if and only if

F F (x)has no any real critical points; F F (+∞) = 0,F F (−∞) = A. Or,

F F (+∞) = A,F F (−∞) = 0,Acan be a non-zero constant and also can be

the positive or negative inﬁnity.

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The critical points of entire fractions and B.Shapiro’s 12th conjecture of entire functions

We give example 1.23 to show that case described in Theorem 1.22 does

occur.

Example 2.23. f8(x) = ex,κ>1.

f′

8=ex.f8has no real zeros and critical points. F F8=(f

′

8)1

κ

f8=ex

κ

ex=

e(1

κ−1)x.f′′

8=ex.f8f′′

8−κ(f′

8)2= (1 −κ)e2x.κ>1. F F8(+∞) = 0.

F F8(−∞) = +∞.κ<1. F F7(+∞) = +∞.F F7(−∞) = 0. It is contrast to

Shapiro’s conjecture.

Applying Lemma 1.15, Lemma 1.17, Lemma 1.19, Lemma 1.21 and

Theorem 1.22 we get Theorem 2.

Declarations

•Acknowledgements The authors would like to thank the reviewers for valu-

able comments and suggestions that led to improvements of the exposition

of the paper.

•Ethical Approval There are no human or animal participates in the study.

•Availability of data and materials Data sharing not applicable to this paper

as no datasets were generated or analyzed during the current study.

•Funding Funding information is not applicable/No funding was received.

•Author Contributions The two authors contributed equally to this paper.

•Competing interests The authors declare that they have no conﬂict of

interest.

References

[1] E.Laguerre, Oeuvres de Laguerre. Tome I, pp. 167–180. Gauthiers-Villars,

Paris, 1898. Reprinted by Chelsea, Bronx, NY, 1972.

[2] E.Borel, Le¸cons sur les Fonctions Enti`eres. Gauthier-Villars, Paris, 1900.

[3] C. F. Gauss, Werke 3 (1886), 119-121.

[4] N. Obreschkoﬀ, Verteilung und Berechnung der Nullstellen releer Poly-

nome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.

[5] D. Nicks, Non-real zeroes of real entire derivatives, J. Anal. Math. 117

(2012), 87-118.

[6] W. Bergweiler, A. Eremenko, and J. K. Langley, Real entire functions of

inﬁnite order and a conjecture of Wiman, Geom. Funct. Anal. 13 (2003),

975-991.

[7] W. Bergweiler and A. Eremenko, Proof of a conjecture of P´olya on the

zeros of successive derivatives of real entire functions, Acta Math. 197

Springer Nature 2021 L

A

T

EX template

10 The critical points of entire fractions and B.Shapiro’s 12th conjecture of entire functions

(2006), 145-166.

[8] J.K. Langley, Non-real zeros of derivatives of real meromorphic functions

of inﬁnite order, Math. Proc. Camb. Philos. Soc. 150 (2011), 343–351.

[9] G. Csordas, T. Craven, W. Smith, The zeros of derivatives of entire

functions and the Wiman-P´olya conjecture, Ann. Math. 125 (2) (1987),

405–431.

[10] G. Csordas, Linear operators and the distribution of zeros of entire

functions, Complex Var. Elliptic Equ. 51 (7) (2006), 625–632.

[11] M. Tyaglov, On the number of critical points of logarithmic derivatives

and the Hawaii conjecture, J. Anal. Math. 114 (2011), 1-62.

[12] B. Shapiro, Problems around polynomials: the good, the bad, and the

ugly . . ., Arnold Math. J. 1 (1) (2015), 91–99.