8. Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1 Issue 3(2009), Pages 85-89.

COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH HANKEL

DETERMINANT

(DEDICATED IN OCCASION OF THE 65-YEARS OF PROFESSOR R.K. RAINA)

GANGADHARAN. MURUGUSUNDARAMOORTHY AND NANJUNDAN. MAGESH

Abstract. In this paper we obtain the functional∣𝑎2𝑎4−𝑎23∣for the class

𝑓∈𝑅(𝛼). Also we give sharp upper bound for∣𝑎2𝑎4−𝑎23∣.Our result extends corresponding previously known result.

1. _{Introduction and Preliminaries}

Let𝐴denote the class of functions of the form

𝑓(𝑧) =𝑧+

∞

∑

𝑛=2

𝑎𝑛𝑧𝑛 (1.1)

which are analytic and univalent in the unit disc 𝑈 ={𝑧: ∣𝑧∣<1}. Let𝑃 be the family of all functions𝑝analytic in 𝑈 for which𝑅𝑒{𝑝(𝑧)}>0 and

𝑝(𝑧) = 1 +

∞

∑

𝑛=1

𝑐𝑛𝑧𝑛, 𝑧∈𝑈. (1.2)

In 1976, Noonan and Thomas [10] deﬁned the𝑞th Hankel determinant of𝑓 for 𝑞≥1 by

𝐻𝑞(𝑛) =

𝑎𝑛 𝑎𝑛+1 . . . 𝑎𝑛+𝑞−1 𝑎𝑛+1 𝑎𝑛+2 . . . 𝑎𝑛+𝑞

..

. ... ... ... 𝑎𝑛+𝑞−1 𝑎𝑛+𝑞 . . . 𝑎𝑛+2𝑞−2

.

Further, Fekete and Szeg¨o [1] considered the Hankel determinant of𝑓 ∈𝐴 for

𝑞= 2 and𝑛= 1, 𝐻2(1) =

𝑎1 𝑎2 𝑎2 𝑎3

.They made an early study for the estimates

2000Mathematics Subject Classiﬁcation. 30C45.

Key words and phrases. Hankel determinant, Fekete-Szeg¨o functional, positive real functions. c

⃝2009 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted October, 2009. Published November, 2009.

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of ∣𝑎3−𝜇𝑎22∣when 𝑎1= 1 with 𝜇real. The well known result due to them states that if𝑓 ∈𝐴, then

∣𝑎3−𝜇𝑎22∣ ≤ ⎧

⎨

⎩

4𝜇−3 if 𝜇≥1,

1 + 2 𝑒𝑥𝑝(−₁₋2𝜇_𝜇) if 0≤𝜇≤1,

3−4𝜇 if 𝜇≤0.

Furthermore, Hummel [3, 4] obtained sharp estimates for ∣𝑎3−𝜇𝑎22∣ when 𝑓 is convex functions and also Keogh and Merkes [6] obtained sharp estimates for ∣𝑎3−𝜇𝑎22∣when𝑓 is close-to-convex, starlike and convex in𝑈.

Here we consider the Hankel determinant of𝑓 ∈𝐴for𝑞= 2 and 𝑛= 2,

𝐻2(2) =

𝑎2 𝑎3 𝑎3 𝑎4

.

In the present investigation we consider the following subclass𝑅(𝛼) of𝐴:

𝑅(𝛼) = {

𝑓(𝑧)∈𝐴: 𝑅𝑒

{

(1−𝛼)𝑓(𝑧) 𝑧 +𝛼𝑓

′₍_𝑧₎}_>₀_{, 𝛼 >}₀_{, 𝑧}_∈_𝑈} _(1.3)

and obtain sharp upper bound for the functional∣𝑎2𝑎4−𝑎32∣of𝑓 ∈𝑅(𝛼).

Remark. The subclass 𝑅(1) = 𝑅 was studied systematically by MacGregor [9]

who indeed referred to numerous earlier investigations involving functions whose derivative has a positive real part.

To prove our main result, we need the following lemmas.

Lemma 1.1. [11] If𝑝∈𝑃, then∣𝑐𝑘∣ ≤2 for each𝑘.

Lemma 1.2. [2] The power series for 𝑝(𝑧) given in (1.2) converges in 𝑈 to a function in𝑃 if and only if the Toeplitz determinants

𝐷𝑛=

2 𝑐1 𝑐2 ⋅ ⋅ ⋅ 𝑐𝑛 𝑐−1 2 𝑐1 ⋅ ⋅ ⋅ 𝑐𝑛−1

..

. ... ... ... ... 𝑐−𝑛 𝑐−𝑛+1 𝑐−𝑛+2 ⋅ ⋅ ⋅ 2

, 𝑛= 1,2,3, . . . (1.4)

and𝑐−𝑘=𝑐𝑘,are all nonnegative. They are strictly positive except for

𝑝(𝑧) = 𝑚 ∑

𝑘=1

𝜌𝑘𝑝0(𝑒𝑖𝑡𝑘𝑧), 𝜌𝑘 >0, 𝑡𝑘 real

and𝑡𝑘 ∕=𝑡𝑗 for𝑘∕=𝑗 in this case𝐷𝑛>0for𝑛 < 𝑚−1and𝐷𝑛= 0 for𝑛≥𝑚.

2. _{Main Result}

Using the techniques of Libera and Zlotkiewicz [7, 8], we now prove the following theorem.

Theorem 2.1. Let 𝛼 >0.If 𝑓 ∈𝑅(𝛼), then

∣𝑎2𝑎4−𝑎23∣ ≤ 4

(1 + 2𝛼)2. (2.1)

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Proof. Since𝑓 ∈𝑅(𝛼),it follows from (1.3) that

(1−𝛼)𝑓(𝑧) 𝑧 +𝛼𝑓

′₍_𝑧_{) =}_𝑝₍_𝑧₎ _(2.2)

for some𝑝∈𝑃.Equating coeﬃcients in (2.2), we have,

(1 +𝛼)𝑎2=𝑐1, (1 + 2𝛼)𝑎3=𝑐2, (1 + 3𝛼)𝑎4=𝑐3. (2.3)

From (2.3), it can be established that

∣𝑎2𝑎4−𝑎23∣=

𝑐1𝑐3

(1 +𝛼)(1 + 3𝛼)− 𝑐2

2 (1 + 2𝛼)2

.

We make use of Lemma 1.2 to obtain the proper bound on

𝑐1𝑐3

(1+𝛼)(1+3𝛼)− 𝑐2 2 (1+2𝛼)2 . We may assume without restriction that 𝑐1 >0. We begin by rewriting (1.4) for the cases𝑛= 2 and 𝑛= 3,

𝐷2 =

2 𝑐1 𝑐2 𝑐1 2 𝑐1 𝑐2 𝑐1 2

= 8 + 2Re{𝑐2₁𝑐2} −2∣𝑐2∣2−4𝑐21≥0, (2.4)

which is equivalent to

2𝑐2=𝑐21+𝑥(4−𝑐21) (2.5)

for some𝑥,∣𝑥∣ ≤1.Then𝐷3≥0 is equivalent to

∣(4𝑐3−4𝑐1𝑐2+𝑐31)(4−𝑐 2

1) +𝑐1(2𝑐2−𝑐21)

2 _{∣ ≤}₂₍₄₋_𝑐2 1)

2₋₂_∣₂_𝑐 2−𝑐21∣

2 _(2.6)

and from (2.6) with (2.5), we have,

4𝑐3 = 𝑐31+ 2(4−𝑐 2

1)𝑐1𝑥−𝑐1(4−𝑐21)𝑥

2_{+ 2(4}₋_𝑐2

1)(1− ∣𝑥∣

2₎_𝑧, _(2.7)

for some value of𝑧,∣𝑧∣ ≤1.

Suppose𝑐1=𝑐and𝑐∈[0,2].Using (2.5) along with (2.7) we obtain

𝑐1𝑐3

(1 +𝛼)(1 + 3𝛼)− 𝑐2

2 (1 + 2𝛼)2

=

𝛼2_𝑐4_{+ 2}_𝛼2_𝑐2₍₄₋_𝑐2₎_𝑥₋₍₁₂_𝛼2_{+ 16}_𝛼₊_𝛼2_𝑐2_{+ 4)(4}₋_𝑐2₎_𝑥2 4(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎ +

𝑐(4−𝑐2₎₍₁_{− ∣}_𝑥_∣2₎_𝑧 2(1 +𝛼)(1 + 3𝛼)

≤ 𝛼

2_𝑐4

4(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎+

𝑐(4−𝑐2₎ 2(1 +𝛼)(1 + 3𝛼)+

𝛼2_𝑐2₍₄₋_𝑐2₎_𝜌 2(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎

+(𝑐−2)(4−𝑐

2_)[_𝛼2₍_𝑐₋₆₎₋₈_𝛼₋_2]_𝜌2

4(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎

≡𝐹(𝜌) (2.8)

with𝜌=∣𝑥∣ ≤1 and𝛼 >0.We assume that the upper bound for (2.8) is attained at an interior point of the set{(𝜌, 𝑐)∣𝜌∈[0,1], 𝑐∈[0,2]},then

𝐹′(𝜌) = 𝛼

2_𝑐2₍₄₋_𝑐2₎

2(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎+

(𝑐−2)(4−𝑐2_)[_𝛼2₍_𝑐₋₆₎₋₈_𝛼₋_2]_𝜌 2(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎ . (2.9)

We note that𝐹′₍_𝜌₎_>_{0 and consequently F is increasing and max}_𝐹₍_𝜌_{) =}_𝐹₍₁₎_,

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𝜌∈[0,1].Now let

𝐺(𝑐) =𝐹(1) = 𝛼 2_𝑐4

4(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎+

𝑐(4−𝑐2₎ 2(1 +𝛼)(1 + 3𝛼)+

𝛼2_𝑐2₍₄₋_𝑐2₎ 2(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎

+(𝑐−2)(4−𝑐

2_)[_𝛼2₍_𝑐₋₆₎₋₈_𝛼₋_2]

4(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎

then

𝐺′(𝑐) = −2𝑐[𝛼

2_𝑐2_{+ 4}_𝛼_{+ 1]}

(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎ = 0 (2.10) therefore (2.10) implies𝑐= 0,which is a contradiction. We note that

𝐺′′(𝑐) = −6𝛼

2_𝑐2₋₈_𝛼₋₂

(1 +𝛼)(1 + 2𝛼)2_{(1 + 3}_𝛼₎ <0.

Thus any maximum points of𝐺must be on the boundary of𝑐∈[0,2].However, 𝐺(𝑐)≥𝐺(2) and thus 𝐺has maximum value at𝑐= 0.The upper bound for (2.8) corresponds to𝜌= 1 and𝑐= 0,in which case

𝑐1𝑐3

(1 +𝛼)(1 + 3𝛼)− 𝑐2

2 (1 + 2𝛼)2

≤ 4

(1 + 2𝛼)2, 𝛼 >0.

This completes the proof of the Theorem 2.1. _□

Remark. If 𝛼 = 1, then we get the corresponding functional ∣𝑎2𝑎4−𝑎23∣ for the

class𝑓 ∈𝑅(1) =𝑅,studied in[5]as in the following corollary.

Corollary 2.2. If 𝑓 ∈𝑅,then

∣𝑎2𝑎4−𝑎23∣ ≤ 4 9.

The result is sharp.

Acknowledgments. The authors would like to thank the referee for his valuable comments and suggestions.

References

[1] M.Fekete and G.Szeg¨o, Eine Bemerkung ¨uber ungerade schlichte Funktionen, J. London. Math. Soc.,8(1933), 85–89.

[2] U.Grenander and G.Szeg¨o, Toeplitz forms and their applications,Univ. of California Press, Berkeley and Los Angeles,1958.

[3] J.Hummel, The coeﬃcient regions of starlike functions, Paciﬁc. J. Math., 7(1957), 1381– 1389.

[4] J.Hummel, Extremal problems in the class of starlike functions,Proc. Amer. Math. Soc.,11

(1960), 741–749.

[5] A.Janteng, S.A.Halim and M.Darus,Coeﬃcient inequality for a function whose derivative has a positive real part, J.Ineq. Pure and Appl. Math., Vol.7, 2 (50) (2006), 1–5.

[6] F.R.Keogh and E.P.Merkes, A coeﬃcient inequality for certain classes of analytic functions,

Proc. Amer. Math. Soc.,20(1969), 8–12.

[7] R.J.Libera and E.J.Zlotkiewicz, Early coeﬃcients of the inverse of a regular convex function,

Proc. Amer. Math. Soc.,85(2)(1982), 225–230.

[8] R.J.Libera and E.J.Zlotkiewicz, Coeﬃcient bounds for the inverse of a function with derivative in𝑃,Proc. Amer. Math. Soc.,87(2)(1983), 251–289.

[9] T.H.MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc.,104(1962), 532–537.

[10] J.W.Noonan and D.K.Thomas, On the second Hankel determinant of areally mean𝑝−valent functions,Trans. Amer. Math. Soc.,223 (2)(1976), 337–346.

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Gangadharan. Murugusundaramoorthy

School of Advanced Sciences , VIT University, Vellore - 632014, India.

E-mail address:[email protected] Nanjundan. Magesh

Department of Mthematics, Government of Arts College(Men), Krishnagiri - 635001, India.