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Toeplitz matrices whose elements are coefficients of Bazilevič functions

  • Varadharajan Radhika , Jay M. Jahangiri , Srikandan Sivasubramanian and Gangadharan Murugusundaramoorthy EMAIL logo
Published/Copyright: October 29, 2018
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Open Mathematics
From the journal Open Mathematics Volume 16 Issue 1

Abstract

We consider the Toeplitz matrices whose elements are the coefficients of Bazilevič functions and obtain upper bounds for the first four determinants of these Toeplitz matrices. The results presented here are new and noble and the only prior compatible results are the recent publications by Thomas and Halim [1] for the classes of starlike and close-to-convex functions and Radhika et al. [2] for the class of functions with bounded boundary rotation.

Keywords: Toeplitz Determinants; Analytic; Univalent and Bazilevič Functions
MSC 2010: 30C45; 33C50; 30C80

1 Introduction

Let A denote the class of all functions f of the form

f(z)=z+∑n=2∞anzn,(1)

which are analytic in the open unit disk U = {z:|z| < 1} and let S denote the subclass of A consisting of univalent functions. Obviously, for functions f ∈ S we must have f′ ≠ 0, in U. For f ∈ S, we consider the family B(β) of Bazilevič functions of type β; 0 ≤ β ≤ 1 so that

ℜz1−βf′(z)f(z)1−β>0.

The family B(β) of Bazilevič functions of type β; 0 ≤ β ≤ 1 provides a transition from the class of starlike functions to the class of functions of bounded boundary rotation. To see this, we note that for the choice of β = 0, we have B(β) ≡ S* (0) ≡ S*, the class of starlike functions f ∈ S so that R(zf′/f) > 0 in and for the choice of β = 1, we get the family R of functions f ∈ S of bounded boundary rotation so that R(f′) > 0 in U. (For further details see [3].)

Several authors (e.g. see [4-8]) have discussed various subfamilies of the well-known Bazilevič functions of type β from various viewpoints including their coefficient estimates. It is interesting to note in this connection that the earlier investigations on the subject do not seem to have made use of Toeplitz matrices and determinants. Toeplitz matrices are one of the well-studied classes of structured matrices. They arise in all branches of pure and applied mathematics, statistics and probability, image processing, quantum mechanics, queueing networks, signal processing and time series analysis, to name a few (e.g see Ye and Lim [9]). Toeplitz matrices have some of the most attractive computational properties and are amenable to a wide range of disparate algorithms and determinant computations.

Here we consider the symmetric Toeplitz determinant

Tq(n)=anan+1⋯an+q−1an+1⋯⋮⋮an+q−1⋯an

and obtain upper bounds for the coefficient body Tq(n); q = 2,3; n = 1, 2, 3 where the entries of Tq(n) are the coefficients of functions of the form (1) that are in the family of Bazilevič functions B(β). As far as we are concerned, the results presented here are new and noble and the only prior compatible results are the recent publications by Thomas and Halim [1] for the classes of starlike and close-to-convex functions and Radhika et al. [2] for the class of functions with bounded boundary rotation. We shall need the following result [10] in order to prove our main theorems.

Lemma 1.1

Leth(z)=1+∑n=1∞pnzn ∈ P. Then for some complex valuedxwith |x| ≤ 1 and some complex valuedζwith |ζ| ≤ 1.

2p2=p12+x(4−p12)4p3=p13+2(4−p12)p1x−p1(4−p12)x2+2(4−p12)(1−|x|2)ζ.

2 Coefficient estimates for Toeplitz determinant

In our first theorem we determine a sharp upper bound for the coefficient body T2(2).

Theorem 2.1

Let f given by (1) be in the classB(β); 0 ≤ β ≤ 1. Then we have the sharp bound

|T2(2)|=a32−a22 ≤ 4(β+2)2max1,(−β4−6β3−12β2−6β+5)(β+1)4.

Proof

First note that by equating the corresponding coefficients in the equation

z1−βf′(z)f(z)1−β=h(z)

we obtain

a2=p1(β+1),(2)
a3=p2β+2−(β−1)p122(β+1)2.(3)

In view of (2) and (3), a simple computation leads to

a32−a22=p22(β+2)2+(β−1)2p144(β+1)4−p12p2(β−1)(β+2)(β+1)2−p12(β+1)2.(4)

Note that, by Lemma 1.1, we may write 2p2 = p2 + x(4 − p2) where without loss of generality we let 0 ≤ p1 = p ≤ 2. Substituting this into the above equation we obtain the following quadratic equation in terms of x.

a32−a22=(4−p2)24(β+2)2x2+(β+3)(4−p2)p22(β+1)2(β+2)2x+(β2+6β+9)p2−4(β+1)2(2+β)2p24(β+1)4(β+2)2.

Using the triangle inequality we obtain

a32−a22 ≤ (4−p2)24(β+2)2+(β+3)(4−p2)p22(β+1)2(β+2)2+(β2+6β+9)p2+4(β+1)2(2+β)2p24(β+1)4(β+2)2=Φ(p,β).

Differentiating Φ(p,β) with respect to p we obtain

∂Φ(p,β)∂p=pp2β3−3β+2−2β3+4β2+14β+8(β+1)4(β+2).

Setting ∂Φ(p,β)∂p=0 yields either p = 0 or

p2=2β3−4β2−14β−8β3−3β+2.

But 2β3 − 4β2 − 14β − 8 < 0 for 0 ≤ β ≤ 1. Therefore, the maximum of |a32−a22| is attained at the end points p1 = p ∈ [0,2].

For p1=0, we have p2 = 2x. Therefore, from (4),

a32−a22=4x2(β+2)2 ≤ 4(β+2)2.

For p1=2 we have a2=2β+1 and a3=2β+2−2(β−1)(β+1)2 which yields

a32−a22 ≤ 4(−β4−6β3−12β2−6β+5)(β+1)4(β+2)2.

The result is sharp for the functions given by

z1−βf′(z)f(z)1−β=1+z1−z.

□

Remark 2.2

Theorem 2.1 for β = 0 yields the bound|a32−a22| ≤ 5 for the class of starlike functions confirming the bound obtained by Thomas and Halim [1] and for β = 1 yields the bound|a32−a22| ≤ 5/9 for the class of functions with bounded boundary rotation R confirming the bound obtained by Radhika et al. [2].

In our next theorem, we determine an upper bound for the coefficient body T2(3).

Theorem 2.3

Let f given by (1), be in the class B(β), 0 ≤ β ≤ 1. Then

T2(3)=a42−a32 ≤ max|N(β)|9(β+3)2(β+1)6(β+2)2,4(2+β)2.

where

N(β)=4β8+40β7+152β6+88β5−1312β4−5096β3−8024β2−4248β+2268.

Proof

By equating the corresponding coefficients in the equation,

z1−βf′(z)f(z)1−β=h(z)

we obtain

a2=p1(β+1),(5)
a3=p2β+2−(β−1)p122(β+1)2, and(6)
a4=p3β+3−(β−1)p1p2(β+1)(β+2)+(β−1)(2β−1)p136(β+1)3.(7)

In view of (6) and (7) and applying Lemma 1, denoting X = 4 − p2 and Y = (1 − |x|2)ζ, where 0 ≤ p ≤ 2 and |ζ| < 1 we get,

a42−a32=β8+10β7+47β6+148β5+383β4+778β3+1153β2+1368β+1296144(β+3)2(β+1)6(β+2)2p16−β2+6β+94(β+1)4(β+2)2p14+X2Y24(β+3)2−p1x2X2Y4(β+3)2+(β+5)xX2p1Y2(β+3)2(β+2)(β+1)+β4+5β3+11β2+19β+3612(β+3)2(β+2)(β+1)3p13XY+p12X2x416(β+3)2−β+54(β+3)2(β+2)(β+1)p12x3X2+β2+10β+254(β+3)2(β+2)2(β+1)2p12X2x2−x2X24(β+2)2−β4+5β3+11β2+19β+3624(β+3)2(β+2)(β+1)3Xx2p14+β5+10β4+36β3+74β2+131β+18012(β+3)2(β+2)2(β+1)4Xxp14−(β+3)2(β+2)2(β+1)2Xxp12.

As in the proof of Theorem 1, without loss of generality, we can write p1 = p, where 0 ≤ p ≤ 2. Then an application of triangle inequality gives,

a42−a32≤(2−p)216(β+3)2|x|4+(β+5)(p2−2p)(4−p2)24(β+3)2(β+2)(β+1)|x|3+β2+10β+254(β+3)2(β+2)(β+1)2p2(4−p2)+p3(p−2)(4−p2)(β4+5β3+11β2+19β+36)24(β+3)2(β+2)(β+1)3−(β2+2β−1)(4−p2)24(β+3)2(β+2)2+p(4−p2)24(β+3)2|x|2+(β+3)2(β+2)2(β+1)2(4−p2)p2+(β+5)(4−p2)2p2(β+3)2(β+2)(β+1)+(β5+10β4+36β3+74β2+131β+180)p4(4−p2)12(β+3)2(β+1)4(β+2)2|x|+N1(β)p6−N2(β)p4+(4−p2)24(β+3)2+(β4+5β3+11β2+19β+36)p3(4−p2)12(β+3)2(β+2)(β+1)3=Ψ(p,|x|)

where

N1(β)=β8+10β7+47β6+148β5+383β4+778β3+1153β2+1368β+1296144(β+3)2(β+2)2(β+1)6,N2(β)=β2+6β+94(β+2)2(β+1)4.

We need to find the maximum value of Ψ(p, |x|) on [0,2] × [0,1]. First, assume that there is a maximum at an interior point Ψ(p0, |x0|) of [0,2] × [0,1]. Differentiating Ψ(p, |x|) with respect to |x| and equating it to 0 implies that p = p0 = 2, which is a contradiction. Thus for the maximum of Ψ(p, |x|), we need only to consider the end points of [0,2] × [0,1].

For p = 0 we obtain

Ψ(0,|x|)=4(β+3)2|x|4−4(β2+2β−1)(β+3)2(β+2)2|x|2+4(β+3)2 ≤ 4(β+2)2.

For p=2 we obtain

Φ(2,|x|)=64N1(β)−16N2(β).

For |x|=0 we obtain

Ψ(p,0)=N1(β)p6−N2(β)p4+(β4+5β3+11β2+19β+36)p3(4−p2)12(β+3)2(β+2)(β+1)3+p(4−p2)24(β+3)2,

which has the maximum value |N1(β)p6 − N2(β)p4| on [0,2].

For |x| = 1 we obtain

Ψ(p,1)=(2−p)2(4−p2)216(β+3)2+(β+5)(p2−2p)(4−p2)24(β+3)2(β+2)(β+1)+(β2+10β+25)p2(4−p2)24(β+3)2(β+2)(β+1)2+p(4−p2)24(β+3)2+(β4+5β3+11β2+19β+36)p3(p−2)(4−p2)24(β+3)2(β+2)(β+1)3−(β2+2β−1)(4−p2)24(β+3)2(β+2)2+(β+3)2(β+2)2(β+1)2(4−p2)p2+(β+5)(4−p2)2p2(β+3)2(β+2)(β+1)+(β5+10β4+36β3+74β2+131β+180)p4(4−p2)12(β+3)2(β+2)2(β+1)4+N1(β)p6−N2(β)p4+(4−p2)24(β+3)2+(β4+5β3+11β2+19β+36)p3(4−p2)12(β+3)2(β+2)(β+1)3

which has the maximum values |64N1(β)−16N2(β)| for p = 2 and 4(2+β)2 for p = 0. □

Remark 2.4

Theorem 2 for β = 0 yields the bound |T2(3)| ≤ 7 for the class of starlike functions S* confirming the bound obtained by Thomas and Halim [1] and for β = 1 yields the bound |T2(3)| ≤ 4/9 for the class of functions with bounded boundary rotation R confirming the bound obtained by Radhika et al. [2].

Theorem 2.5

Let f given by (1) be in the class B(β), (0 ≤ β ≤ 1; β ≠ β0), then

T3(2)=a2a3a4a3a2a3a4a3a2≤max|M1(β)M2(β)|, 8 |M1(β)|(β+2)2;if β≠β0,max|M2(β)M3(β)|, 8 |M3(β)|(β+2)2;if β=β0,

where β0 ≈ 0.3676 is the positive root of the polynomial

4β4+32β3+80β2+64β−36=0,
M1(β)=4β4+32β3+80β2+64β−363(β+1)3(β+2)(β+3),M2(β)=4(4β5+34β4+108β3+140β2+32β−54)3(β+1)4(β+2)2(β+3)

and

M3(β)=8β4+52β3+124β2+140β+1083(β+1)3(β+2)(β+3).

Proof

Write

T3(2)=a23−2a2a32+2a32a4−a2a4=(a2−a4)(a22−2a32+a2a4).

Using the same techniques as in Theorem 2, one can obtain with simple computations that

|a2−a4| ≤ M1(β)  for  β≠β0.

We need to show that

a22−2a32+a2a4 ≤ |M2(β)|.

In view of (2), (3) and (7) and Lemma 1, where we denote X = 4 − p2 and Y = (1 − |x|2)ζ, where 0 ≤ p ≤ 2 and |ζ| < 1, one may easily get,

a22−2a32+a2a4=p12(β+1)2−2p22(β+2)2+(β−1)2p144(β+1)4−(β−1)p12p2(β+1)2(β+2)+p1(β+1)p3β+3−(β−1)p1p2(β+1)(β+2)+(β−1)(2β−1)p136(β+1)3=p12(β+1)2−2p22(β+2)2−(β−1)2p142(β+1)4+2(β−1)p12p2(β+1)2(β+2)+p1p3(β+1)(β+3)−(β−1)p12p2(β+1)2(β+2)+(β−1)(2β−1)p146(β+1)4=p12(β+1)2−12(β+2)2p14+X2x2+2p12Xx−(β−1)2p142(β+1)4+(β−1)p122(β+1)2(β+2)p12+Xx+p14(β+1)(β+3)p13+2p1Xx−p1Xx2+2XY+(β−1)(2β−1)p146(β+1)4.

Applying the triangle inequality and assuming that p1 = p, where 0 ≤ p ≤ 2 we obtain

a22−2a32+a2a4 ≤ 14(β+1)(β+3)p2−2p+12(2+β)2(4−p2)(4−p2)|x|2+(β2+5β+8)2(β+1)2(β+2)2(β+3)p2(4−p2)|x|+p2(β+1)2−(90−β5−7β4−15β3+13β2+88β)p412(β+1)4(β+2)2(β+3)+p2(β+1)(β+3)(4−p2)=Ω(p,|x|)

We need to find the maximum value of Ω(p,|x|) on [0,2] × [0,1]. First, assume that there is a maximum at an interior point Ω(p0,|x0|) of [0,2] × [0,1]. Differentiating Ω([,|x|) with respect to |x| and equating it to zero implies that p = p0 = 2, which is a contradiction. Thus for the maximum of Ω(p,|x|), we need only to consider the end points of [0,2] × [0,1].

For p = 0 we obtain

Ω(0,|x|)=8(β+2)2|x|2 ≤ 8(β+2)2.

For p = 2 we obtain

Ω(2,|x|)=4(4β5+34β4+108β3+140β2+32β−54)3(β+1)4(β+2)2(β+3)=M2(β).

For |x| = 0 we obtain

Ω(p,0)=p2(β+1)2−(90−β5−7β4−15β3+13β2+88β)12(β+1)4(β+2)2(β+3)p4+p2(β+1)(β+3)(4−p2)

which has maximum value Ω(p,0) = M2(β) attained at the end point p = 2.

For |x| = 1 we obtain

Ω(p,1)=p2(β+1)2−(90−β5−7β4−15β3+13β2+88β)12(β+1)4(β+2)2(β+3)p4+12(β+2)2(4−p2)2+14(β+1)(β+3)p2(4−p2)+(β2+5β+8)2(β+1)2(β+2)2(β+3)p2(4−p2),

which has maximum value Ω(p,1)=8(β+2)2 at p = 0 and Ω(p,1) = M2(β) at p = 2. Hence

a22−2a32+a2a4 ≤ max|M2(β)|,8(β+2)2.

Thus

T3(2)=(a2−a4)(a22−2a32+a2a4) ≤ max|M1(β)M2(β)|,8 |M1(β)|(β+2)2.

For the case β = β0, we compute |a2−a4| as follows

|a2−a4|=p1β+1−p3β+3+(β−1)p1p2(β+1)(β+2)−(β−1)(2β−1)p136(β+1)3.

Since, each |pi| ≤ 2, an application of triangle inequality shows that

|a2−a4| ≤ |M3(β)|=8β4+52β3+124β2+140β+1083(β+1)3(β+2)(β+3).

Therefore

T3(2)=(a2−a4)(a22−2a32+a2a4) ≤ max|M2(β)M3(β)|,8 |M3(β)|(β+2)2.

This completes the proof of Theorem 2.5. □

Remark 2.6

Theorem 2.5 for β = 0 yields the bound |T3(2)| ≤ 8 for the class of starlike functions S* confirming the bound obtained by Thomas and Halim [1] and for β = 1 yields the bound |T3(2)| ≤ 4/9 for the class of functions with bounded boundary rotation R confirming the bound obtained by Radhika et al. [2].

Theorem 2.7

Let f given by (1), be in the class B(β), 0 ≤ β ≤ 1. Then

T3(1)=1a2a3a21a2a3a21 ≤ max1+14(β+2)2,|M4(β)|

where

M4(β)=β6+8β5+18β4−4β3−51β2−20β+32(β+1)4(β+2)2.

Proof

Expanding the determinant by using equations (2) and (3) and applying Lemma 1.1, we have

T3(1)=1+2a22(a3−1)−a32=1+2p12(β+1)2p2(β+2)−(β−1)p122(β+1)2−1−p22(β+2)2−(β−1)2p144(β+1)4+p2p12(β−1)(β+1)2(β+2)=1+2p12(β+1)2(2+β)p12+Xx2−(β−1)p14(β+1)4−2p12(β+1)2−(β−1)2p144(β+1)4−14(β+2)2p14+X2x2+2p12Xx+p12(β−1)(β+1)2(β+2)p12+Xx2=1+p14(β+1)2(β+2)+p12Xx(β+1)2(β+2)−(β−1)p14(β+1)4−2p12(β+1)2−p144(β+2)2−X2x24(β+2)2−p12Xx2(β+2)2−(1−β)2p144(β+1)4−p14(1−β)2(β+1)2(β+2)−p12Xx(1−β)2(β+1)2(β+2)=1+p141(β+1)2(2+β)+(1−β)(β+1)4−14(β+2)2−(1−β)24(β+1)4−(1−β)2(β+1)4(β+2)−2p12(β+1)2+p12Xx(1+β)(β+2)−(β+1)22(β+1)2(β+2)2−X2x24(β+2)2−2p12(β+1)2=1+p144(β+1)4(β+2)24(β+1)2(β+2)+4(1−β)(β+2)2−(β+1)4−(1−β)2(β+2)2−2(1−β)(β+1)2(β+2)−2p12(β+1)2+p12Xx(1+β)(β+2)−(β+1)22(β+1)2(β+2)2−X2x24(β+2)2−2p12(β+1)2=1+3β2+14β+154(β+1)4(β+2)2p14−2p12(β+1)2+12(β+1)(β+2)2p12xX−14(β+2)2x2X2.

Without loss of generality, we let 0 ≤ p1 = p ≤ 2. Now substituting this into the above equation and applying the triangle inequality we obtain the following quadratic equation in terms of x.

T3(1) ≤ (4−p2)24(β+2)2x2+p2(4−p2)2(β+1)(β+2)2x+1+(3β2+14β+15)p2−8(β+1)2(β+2)24(β+1)4(β+2)2p2 ≤ (4−p2)24(β+2)2+p2(4−p2)2(β+1)(β+2)2+1+(3β2+14β+15)p2−8(β+1)2(β+2)24(β+1)4(β+2)2p2=Γ(p,β).

Differentiating Γ(p, β) with respect to p we obtain

∂Γ(p,β)∂p=pp2β4+2β3+3β2+12β+14+4β3+13β2+14β+15(β+1)4(β+2)2.

Setting ∂(Γ(p,β))/∂p = 0 yields either p = 0 or

p2=−4β3−13β2−14β−15β4+2β3+3β2+12β+14.

But −4β3 − 13β2 − 14β − 15 < 0 for 0 ≤ β ≤ 1. Therefore, the maximum of |T3(1)| is attained at the end points p1 = p ∈ [0,2].

For p1 = 0 we have a2 = 0 and a3=1−x24(β+2)2 which yields\newline

T3(1)=1+x24(β+2)2 ≤ 1+14(β+2)2.

For p1 = 2 we obtain

T3(1) ≤ 1+4(3β2+14β+15)(β+1)4(β+2)2−8(β+1)2 ≤ M4(β).

where

M4(β)=β6+8β5+18β4−4β3−51β2−20β+32(β+1)4(β+2)2.

This completes the proof of Theorem 2.7. □

Remark 2.8

Theorem 2.5 for β = 0 yields the bound |T3(1)| ≤ 8 for the class of starlike functions S* confirming the bound obtained by Thomas and Halim [1] and for β = 1 yields the bound |T3(1)| ≤ 13/9 for the class of functions with bounded boundary rotation R confirming the bound obtained by Radhika et al. [2].

  1. Conflict of interest

    The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgement

The authors sincerely thank the referees for their insightful suggestions. The work of the third author is supported by a grant from Department of Science and Technology, Government of India vide ref: SR/FTP/MS-022/2012 under fast track scheme.

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Received: 2016-11-15
Accepted: 2018-02-07
Published Online: 2018-10-29

© 2018 Radhika et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

  1. Regular Articles
  2. Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
  5. Singular Cauchy problem for the general Euler-Poisson-Darboux equation
  6. Ternary and n-ary f-distributive structures
  7. On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
  8. Evaluation of integrals with hypergeometric and logarithmic functions
  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
  14. Monotone subsequence via ultrapower
  15. Note on group irregularity strength of disconnected graphs
  16. On the security of the Courtois-Finiasz-Sendrier signature
  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
https://doi.org/10.1515/math-2018-0093
Keywords for this article
Toeplitz Determinants; Analytic; Univalent and Bazilevič Functions
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Articles
  2. Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
  3. On a viscous two-fluid channel flow including evaporation
  4. Generation of pseudo-random numbers with the use of inverse chaotic transformation
  5. Singular Cauchy problem for the general Euler-Poisson-Darboux equation
  6. Ternary and n-ary f-distributive structures
  7. On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
  8. Evaluation of integrals with hypergeometric and logarithmic functions
  9. Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
  10. Oscillation of first order linear differential equations with several non-monotone delays
  11. Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
  12. The log-concavity of the q-derangement numbers of type B
  13. Generalized state maps and states on pseudo equality algebras
  14. Monotone subsequence via ultrapower
  15. Note on group irregularity strength of disconnected graphs
  16. On the security of the Courtois-Finiasz-Sendrier signature
  17. A further study on ordered regular equivalence relations in ordered semihypergroups
  18. On the structure vector field of a real hypersurface in complex quadric
  19. Rank relations between a {0, 1}-matrix and its complement
  20. Lie n superderivations and generalized Lie n superderivations of superalgebras
  21. Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
  22. Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
  23. On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the εm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of Szász-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
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