Integral means inequalities are obtained for the fractional derivatives of order p+λ(0 5 p 5 n

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http://jipam.vu.edu.au/

Volume 3, Issue 5, Article 66, 2002

INTEGRAL MEANS INEQUALITIES FOR FRACTIONAL DERIVATIVES OF SOME GENERAL SUBCLASSES OF ANALYTIC FUNCTIONS

TADAYUKI SEKINE, KAZUYUKI TSURUMI, SHIGEYOSHI OWA, AND H.M. SRIVASTAVA COLLEGE OFPHARMACY

NIHONUNIVERSITY

7-1 NARASHINODAI7-CHOME, FUNABASHI-SHI

CHIBA274-8555, JAPAN

[email protected] DEPARTMENT OFMATHEMATICS

TOKYODENKIUNIVERSITY

2-2 NISIKI-CHO, KANDA, CHIYODA-KU

TOKYO101-8457, JAPAN

[email protected] DEPARTMENT OFMATHEMATICS

KINKIUNIVERSITY

HIGASHI-OSAKA

OSAKA577-8502, JAPAN

[email protected]

DEPARTMENT OFMATHEMATICS ANDSTATISTICS

UNIVERSITY OFVICTORIA

VICTORIA, BRITISHCOLUMBIAV8W 3P4 CANADA

[email protected]

Received 26 June, 2002; accepted 4 July, 2002 Communicated by D.D. Bainov

ABSTRACT. Integral means inequalities are obtained for the fractional derivatives of order p+λ(0 5 p 5 n; 0 5 λ < 1)of functions belonging to certain general subclasses of analytic functions. Relevant connections with various known integral means inequalities are also pointed out.

Key words and phrases: Integral means inequalities, Fractional derivatives, Analytic functions, Univalent functions, Extreme points, Subordination.

2000 Mathematics Subject Classification. Primary 30C45; Secondary 26A33, 30C80.

ISSN (electronic): 1443-5756 c

The present investigation was initiated during the fourth-named author’s visit to Saga National University in Japan in April 2002. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

072-02

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1. INTRODUCTION, DEFINITIONS, ANDPRELIMINARIES

LetAdenote the class of functionsf(z)normalized by f(z) = z+

∞

X

k=2

a_kz^k

that are analytic in the open unit disk

U:={z :z ∈C and |z|<1}.

Also letA(n)denote the subclass ofAconsisting of all functionsf(z)of the form:

f(z) = z−

∞

X

k=n+1

a_kz^k (a_k =0; n ∈N:={1,2,3, . . .}).

We denote byT (n)the subclass ofA(n)of functions which are univalent inU, and byT_α(n) and C_α(n) the subclasses ofT (n)consisting of functions which are, respectively, starlike of orderα(05α <1)and convex of orderα(05α <1)inU. The classesA(n), T(n), T_α(n), and C_α(n) were investigated by Chatterjea [1] (and Srivastava et al. [9]). In particular, the following subclasses:

T :=T(1), T^∗(α) :=T_α(1), and C(α) :=C_α(1) were considered earlier by Silverman [7].

Next, following the work of Sekine and Owa [4], we denote byA(n, ϑ) the subclass ofA consisting of all functionsf(z)of the form:

(1.1) f(z) = z−

∞

X

k=n+1

e^i(k−1)ϑa_kz^k (ϑ∈R; a_k =0; n∈N).

Finally, the subclassesT(n, ϑ), T_α^∗(n, ϑ), andC_α(n, ϑ)of the classA(n, ϑ)are defined in the same way as the subclassesT(n), T_α(n), andC_α(n)of the classA(n).

We begin by recalling the following useful characterizations of the function classesT_α^∗(n, ϑ) andCα(n, ϑ)(see Sekine and Owa [4]).

Lemma 1.1. A functionf(z)∈ A(n, ϑ)of the form(1.1)is in the classT_α^∗(n, ϑ)if and only if (1.2)

∞

X

k=n+1

(k−α) a_k51−α (n∈N; 05α <1).

Lemma 1.2. A functionf(z)∈ A(n, ϑ)of the form(1.1)is in the classC_α(n, ϑ)if and only if (1.3)

∞

X

k=n+1

k(k−α) a_k 51−α (n ∈N; 0 5α <1).

Motivated by the equalities in (1.2) and (1.3) above, Sekine et al. [6] defined a general sub- class A(n;B_k, ϑ)of the class A(n, ϑ)consisting of functions f(z) of the form (1.1), which satisfy the following inequality:

∞

X

k=n+1

B_ka_k51 (B_k >0; n∈N).

Thus it is easy to verify each of the following classifications and relationships:

A(n;k, ϑ) =T₀^∗(n, ϑ) =:T^∗(n, ϑ) = T (n, ϑ),

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A

n;k−α 1−α, ϑ

=T_α^∗(n, ϑ) (05α <1), and

A

n;k(k−α) 1−α , ϑ

=Cα(n, ϑ) (05α <1).

As a matter of fact, Sekine et al. [6] also obtained each of the following basic properties of the general classesA(n;B_k, ϑ).

Theorem 1.3. A(n;B_k, ϑ)is the convex subfamily of the classA(n, ϑ). Theorem 1.4. Let

f₁(z) = z and f_k(z) = z− e^i(k−1)ϑ B_k z^k (1.4)

(k =n+ 1, n+ 2, n+ 3, . . .; n∈N). Thenf ∈ A(n;B_k, ϑ)if and only iff(z)can be expressed as follows:

f(z) =λ₁f₁(z) +

∞

X

k=n+1

λ_kf_k(z), where

λ1+

∞

X

k=n+1

λk= 1 (λ1 =0; λk =0; n∈N).

Corollary 1.5. The extreme points of the classA(n;B_k, ϑ)are the functionsf₁(z)andf_k(z) (k =n+ 1; n∈N)given by(1.4).

Applying the concepts of extreme points, fractional calculus, and subordination, Sekine et al.

[6] obtained several integral means inequalities for higher-order fractional derivatives and fractional integrals of functions belonging to the general classesA(n;Bk, ϑ). Subsequently, Sekine and Owa [5] discussed the weakening of the hypotheses forBk in those results by Sekine et al.

[6]. In this paper, we investigate the integral means inequalities for the fractional derivatives of f(z) of a general order p+λ (0 5 p 5 n; 0 5 λ < 1) of functionsf(z) belonging to the general classesA(n;B_k, ϑ).

We shall make use of the following definitions of fractional derivatives (cf. Owa [3]; see also Srivastava and Owa [8]).

Definition 1.1. The fractional derivative of orderλis defined, for a functionf(z), by

(1.5) D_z^λf(z) := 1

Γ (1−λ) d dz

Z z

0

f(ζ)

(z−ζ)^λ dζ (05λ <1),

where the functionf(z)is analytic in a simply-connected region of the complexz-plane con- taining the origin and the multiplicity of (z−ζ)^−λ is removed by requiring log (z−ζ) to be real whenz−ζ >0.

Definition 1.2. Under the hypotheses of Definition 1.1, the fractional derivative of ordern+λ is defined, for a functionf(z), by

D_z^n+λf(z) := dⁿ

dzⁿD^λ_zf(z) (05λ <1; n∈N⁰ :=N∪ {0}).

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It readily follows from (1.5) in Definition 1.1 that

(1.6) D_z^λz^k= Γ (k+ 1)

Γ (k−λ+ 1)z^k−λ (05λ <1).

We shall also need the concept of subordination between analytic functions and a subordination theorem of Littlewood [2] in our investigation.

Given two functionsf(z)andg(z), which are analytic inU, the functionf(z)is said to be subordinate tog(z)inUif there exists a functionw(z), analytic inUwith

w(0) = 0 and |w(z)|<1 (z ∈U), such that

f(z) = g(w(z)) (z ∈U). We denote this subordination by

f(z)≺g(z).

Theorem 1.6 (Littlewood [2]). If the functionsf(z)andg(z)are analytic inUwith g(z)≺f(z),

then

Z 2π

0

g re^iθ

µdθ 5 Z 2π

0

f re^iθ

µdθ (µ >0; 0< r <1).

2. THEMAININTEGRAL MEANSINEQUALITIES

Theorem 2.1. Suppose thatf(z)∈ A(n;k^p+1Bk, ϑ)and that (h+ 1)^qB_h+1Γ(h+ 2−λ−p)

Γ(h+ 1) · Γ(n+ 1−p)

Γ(n+ 2−λ−p) 5B_k (k=n+ 1)

for someh=n,05λ <1, and05q5p5n. Also let the functionfh+1(z)be defined by (2.1) f_h+1(z) =z− e^ihϑ

(h+ 1)^q+1Bh+1

z^h+1 f_h+1 ∈A n;k^q+1B_k, ϑ .

Then, forz =re^iθ and 0< r <1, (2.2)

Z 2π

0

D_z^p+λf(z)

µdθ 5 Z 2π

0

D^p+λ_z f_h+1(z)

µdθ (05λ <1; µ >0).

Proof. By virtue of the fractional derivative formula (1.6) and Definition 1.2, we find from (1.1) that

D^p+λ_z f(z) = z^1−λ−p

Γ (2−λ−p) 1−

∞

X

k=n+1

e^i(k−1)ϑ Γ(2−λ−p)Γ(k+ 1)

Γ(k+ 1−λ−p) a_kz^k−1

!

= z^1−λ−p

Γ (2−λ−p) 1−

∞

X

k=n+1

e^i(k−1)ϑΓ(2−λ−p) k!

(k−p−1)!Φ(k)a_kz^k−1

! ,

where

(2.3) Φ(k) := Γ (k−p)

Γ (k+ 1−λ−p) (05λ <1; k=n+ 1; n∈N).

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SinceΦ(k)is a decreasing function ofk, we have

0<Φ (k)5Φ (n+ 1) = Γ (n+ 1−p) Γ(n+ 2−λ−p) (05λ <1; k =n+ 1; n∈N).

Similarly, from (2.1), (1.6), and Definition 1.2, we obtain, for05λ <1, D^p+λ_z f_h+1(z) = z^1−λ−p

Γ (2−λ−p)

1− e^ihϑ

(h+ 1)^q+1B_h+1 · Γ(2−λ−p)Γ(h+ 2) Γ(h+ 2−λ−p) z^h

. Forz =re^iθ and0< r <1, we must show that

Z 2π

0

1−

∞

X

k=n+1

µ

dθ

5 Z 2π

0

1− e^ihϑ

(h+ 1)^q+1B_h+1 · Γ(2−λ−p)Γ(h+ 2) Γ(h+ 2−λ−p) z^h

µ

dθ, (05λ <1; µ >0).

Thus, by applying Theorem 1.6, it would suffice to show that (2.4) 1−

∞

X

k=n+1

≺1− e^ihϑ (h+ 1)^q+1Bh+1

· Γ(2−λ−p)Γ(h+ 2) Γ(h+ 2−λ−p) z^h. Indeed, by setting

1−

∞

X

k=n+1

(k−p−1)! Φ(k)a_kz^k−1

= 1− e^ihϑ

(h+ 1)^q+1B_h+1 · Γ(2−λ−p)Γ(h+ 2)

Γ(h+ 2−λ−p) {w(z)}^h, we find that

{w(z)}^h = (h+ 1)^q+1B_h+1Γ(h+ 2−λ−p) e^ihϑΓ(h+ 2) ·

∞

X

k=n+1

e^i(k−1)ϑ k!

(k−p−1)!Φ(k)a_kz^k−1, which readily yieldsw(0) = 0.

Therefore, we have

|w(z)|^h

5 (h+ 1)^q+1Bh+1Γ(h+ 2−λ−p) Γ(h+ 2)

∞

X

k=n+1

k!

(k−p−1)! Φ(k)a_k|z|^k−1 5|z|ⁿ(h+ 1)^q+1B_h+1Γ(h+ 2−λ−p)

Γ(h+ 2) ·Φ(n+ 1)

∞

X

k=n+1

k!

(k−p−1)!a_k

=|z|ⁿ(h+ 1)^q+1B_h+1Γ(h+ 2−λ−p)

Γ(h+ 2) · Γ(n+ 1−p) Γ(n+ 2−λ−p)

∞

X

k=n+1

k!

(k−p−1)! a_k

=|z|ⁿ(h+ 1)^qB_h+1Γ(h+ 2−λ−p)

Γ(h+ 1) · Γ(n+ 1−p) Γ(n+ 2−λ−p)

∞

X

k=n+1

k!

(k−p−1)!ak

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5|z|ⁿ

∞

X

k=n+1

k!

(k−p−1)!B_ka_k 5|z|ⁿ

∞

X

k=n+1

k^p+1B_ka_k5|z|ⁿ<1 (n∈N), (2.5)

by means of the hypothesis of Theorem 2.1.

In light of the last inequality in (2.5) above, we have the subordination (2.4), which evidently

proves Theorem 2.1.

3. REMARKS ANDOBSERVATIONS

First of all, in its special case whenp=q= 0, Theorem 2.1 readily yields

Corollary 3.1 (cf. Sekine and Owa [5], Theorem 6). Suppose thatf(z) ∈ A(n;kB_k, ϑ)and that

B_h+1Γ(h+ 2−λ)

Γ(h+ 1) · Γ(n+ 1)

Γ(n+ 2−λ) 5B_k (k =n+ 1; n ∈N) for someh=nand 05λ <1. Also let the functionf_h+1(z)be defined by (3.1) fh+1(z) =z− e^ihϑ

(h+ 1)B_h+1 z^h+1 (fh+1 ∈ A(n;kBk, ϑ)). Then, forz =re^iθ and 0< r <1,

(3.2)

Z 2π

0

D^λ_zf(z)

µdθ 5 Z 2π

0

D^λ_zf_h+1(z)

µdθ (05λ <1; µ >0).

A further consequence of Corollary 3.1 whenh=nwould lead us immediately to Corollary 3.2 below.

Corollary 3.2. Suppose thatf(z)∈ A(n;kB_k, ϑ)and that

(3.3) B_n+1 5B_k (k =n+ 1; n ∈N).

Also let the functionfn+1(z)be defined by f_n+1(z) =z− e^inϑ

(n+ 1)B_n+1 zⁿ⁺¹ (f_h+1 ∈ A(n;kB_k, ϑ)). Then, forz =re^iθ and 0< r <1,

Z 2π

0

D_z^λf(z)

µdθ5 Z 2π

0

D^λ_zf_n+1(z)

µdθ (05λ <1; µ > 0).

The hypothesis (3.3) in Corollary 3.2 is weaker than the corresponding hypothesis in an earlier result of Sekine et al. [6, p. 953, Theorem 6].

Next, forp = 1and q = 0, Theorem 2.1 reduces to an integral means inequality of Sekine and Owa [5, Theorem 7] which, for h = n, yields another result of Sekine et al. [6, p. 953, Theorem 7] under weaker hypothesis as mentioned above.

Finally, by setting p = q = 1 in Theorem 2.1, we obtain a slightly improved version of another integral means inequalities of Sekine and Owa [5, Theorem 8] with respect to the pa- rameterλ (see also Sekine et al. [6, p. 955, Theorem 8] for the case whenh = n, just as we remarked above).

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[2] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London Math. Soc. (2), 23 (1925), 481–519.

[3] S. OWA, On the distortion theorems. I, Kyungpook Math. J., 18 (1978), 53–59.

[4] T. SEKINE AND S. OWA, Coefficient inequalities for certain univalent functions, Math. Inequal.

Appl., 2 (1999), 535–544.

[5] T. SEKINE AND S. OWA, On integral means inequalities for generalized subclasses of analytic functions, in Proceedings of the Third ISAAC Congress, Berlin, August 2001.

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[7] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109–116.

[8] H.M. SRIVASTAVA AND S. OWA (Eds.), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1989.

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