PROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1)

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1 GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk are introduced. The object of the present paper is to derive some properties of integral operators P α and Q α.. Introduction Let A be the class of functions of the form = z + a n z n (.) which are analytic in the open unit disk U = z : z <. cently, Jung, Kim, and Srivastava [] have introduced the following one-parameter families of integral operators: and P α f = P α = Q α f = Q α = J α f = J α = α + z α n=2 z ( 2α log z ) α f(t)dt (α > 0), (.2) zγ(α) t 0 ( α + ) z α ( z t ) α t f(t)dt (.3) z 0 (α > 0, > ) z 0 t α f(t)dt (α > ), (.4) 99 Mathematics Subject Classification. 30C45. Key words and phrases. Analytic functions, logarithmic differentiation, integral operator X/95/ $7.50/0 c 995 Plenum Publishing Corporation

2 536 SHIGEYOSHI OWA where Γ(α) is the familiar Gamma function, and (in general) [ ] [ ] α Γ(α + ) α = Γ(α + )Γ( + ) =. (.5) α For α N =, 2, 3,..., the operators P α, Q α, and J α were considered by Bernardi ([2], [3]). Further, for a real number α >, the operator J α was used by Owa and Srivastava [4], and by Srivastava and Owa ([8], [6]). mark. For A given by (.), Jung, Kim, and Srivastava [] have shown that ( 2 ) αan P α = z + z n (α > 0), (.6) n + n=2 Q α Γ(α + + ) Γ( + n) = z + Γ( + ) Γ(α + + n) a nz n (.7) and J α = z + n=2 (α > 0, > ) n=2 By virtue of (.6) and (.8), we see that ( α + ) a n z n (α > ). (.8) α + n J α = Q α (α > ). (.9) 2. An Application of the Miller Mocanu Lemma To derive some properties of operators, we have to recall here the following lemma due to Miller and Mocanu [7]. Lemma. Let w(u, v) be a complex valued function, w : D C, D C C (C is the complex plane), and let u = u + iu 2 and v = v + iv 2. Suppose that the function w(u, v) satisfies the following conditions: (i) w(u, v) is continuous in D; (ii) (, 0) D and w(, 0) > 0; (iii) w(iu 2, v ) 0 for all (iu 2, v ) D and such that v ( + u 2 2)/2. Let p(z) be regular in U and p(z) = + p z + p 2 z such that (p(z), zp (z)) D for all z U. If w(p(z), zp (z)) > 0 (z U), then p(z) > 0 (z U). Applying the above lemma, we derive

3 PROPERTIES OF CERTAIN INTEGRAL OPERATORS 537 Theorem. If A satisfies for some ( < ), then Proof. Noting that we have P P α > (α > 2; z U) (2.) P α P α > (z U). (2.2) z(p α ) = 2P α F (z) P α (α > ), (2.3) z(p α ) P α Define the function p(z) by with P α P α = 2 P α P α (α > ). (2.4) = γ + ( γ)p(z) (2.5) γ = (2.6) 8 Then p(z) = + p z + p 2 z is analytic in the open unit disk U. Since or we have z(p α ) P α z(p α ) P α = ( γ)zp (z) γ + ( γ)p(z), (2.7) P P α = P α ( γ)zp (z) P α + 2γ + ( γ)p(z), (2.8) = γ + ( γ)p(z) + P P α = Therefore, if we define the function w(u, v) by then we see that w(u, v) = γ + ( γ)u + ( γ)zp (z) 2γ + ( γ)p(z) > 0. (2.9) ( γ)v 2γ + ( γ)u, (2.0)

4 538 SHIGEYOSHI OWA ( (i) w(u, v) is continuous in D = C γ γ ) C; (ii) (, 0) D and w(, 0) = > 0; (iii) for all (iu 2, v ) D and such that v ( + u 2 2)/2, w(iu 2, v ) = γ + γ( γ)v 2γ 2 + ( γ) 2 u 2 2 γ γ( γ)( + u2 2) 4γ 2 + ( γ) 2 u 2 2 = ( γ)(γ + 4( γ)) = 4γ 2 + ( γ) 2 u2 2 u This implies that the function w(u, v) satisfies the conditions in Lemma. Thus, applying Lemma, we conclude that P α P α > γ = = Taking the special values for in Theorem, we have Corollary. Let be in the class A. Then P (i) P α > (z U) 2 P α = P α > (z U), 4 P (ii) P α > (z U) 4 P α 5 = P α > (z U), 4 P (iii) P α > 0 (z U) P α 7 = P α > (z U), 8 P (iv) P α > (z U) 4 P α = P α > (z U), 2 and P (v) P α > (z U) 2 P α 7 + = P α > (z U). 8 (z U). (2.)

5 Next, we have PROPERTIES OF CERTAIN INTEGRAL OPERATORS 539 Theorem 2. If A satisfies > γ (α > 2, > ; z U) (2.2) for some γ ((α + 3)/2(α + ) γ < ), then where α Q α > δ (z U), (2.3) δ = + 2γ(α + ) + ( + 2γ(α + )) 2 + 8(α + ). (2.4) 4(α + ) Proof. By the definition of Q α, we know that z(q α ) = (α + ) (α + )Q α (2.5) (α >, > ), so that z(q α ) Q α = (α + ) Qα Q α (α + ). (2.6) We define the function p(z) by Q α = δ + ( δ)p(z). (2.7) Then p(z) = + p z + p 2 z is analytic in U. Making use of the logarithmic differentiations in both sides of (2.7), we have z( ) Applying (2.5) to (2.8), we obtain that Q = + ( δ)zp (z) δ + ( δ)p(z) = z(qα ) Q α + ( δ)zp (z) δ + ( δ)p(z). (2.8) α + = +(α + )( δ)p(z) + (α + ) Qα Q α + α + δ(α + ) + ( δ)zp (z) δ + ( δ)p(z), (2.9)

6 540 SHIGEYOSHI OWA that is, that Now, we let γ = +(α + )( δ)p(z) + w(u, v) = +(α + )( δ)u + α + δ(α + ) + ( δ)zp (z) γ > 0. (2.20) δ + ( δ)p(z) δ(α + ) + α + ( δ)v δ + ( δ)u γ (2.2) with u = u + iu 2 and v = v + iv 2 (. Then w(u, v) satisfies that (i) w(u, v) is continuous in D = C δ δ ) C; (ii) (, 0) D and w(, 0) = γ > 0; (iii) for all (iu 2, v ) D and such that v ( + u 2 2)/2, w(iu 2, v ) = δ(α + ) + α + + δ( δ)v δ 2 + ( δ) 2 u 2 γ δ(α + ) 2 α + γ(α + ) δ( δ)( + u2 2) 2δ 2 + ( δ) 2 u Thus, the function w(u, v) satisfies the conditions in Lemma. This shows that p(z) > 0, or then α Q α > δ (z U). (2.22) If we take γ = (α + 3)/2(α + ) in Theorem 2, then we have Corollary 2. If A satisfies > α + 3 2(α + ) (α > 2, > ; z U), α Q α > 2 (z U). Further, letting α = 2 and γ = /2 in Theorem 2, we have

7 then PROPERTIES OF CERTAIN INTEGRAL OPERATORS 54 Corollary 3. If A satisfies Q Q 2 > 2 ( > ; z U), > (z U). 3. An Application of Jack s Lemma We need the following lemma due to Jack [8], (also, due to Miller and Mocanu [7]). Lemma 2. Let w(z) be analytic in U with w(0) = 0. If w(z) attains its maximum value on the circle z = r < at a point z 0 U, then we can write where k is a real number and k. then z 0 w (z 0 ) = kw(z 0 ), (3.) Applying Lemma 2 for the operator P α, we have Theorem 3. If A satisfies where P P α > (α > 2, /4; z U), (3.2) Proof. Defining the function w(u, v) by P α P α > γ (z U), (3.3) γ = (3.4) 8 P α P α = ( 2γ)w(z), (3.5) + w(z) we see that w(z) is analytic in U and w(0) = 0. It follows from (3.5) that z(p α ) P α = z(p α ) P α ( 2γ)zw (z) ( 2γ)w(z) zw (z) + w(z). (3.6)

8 542 SHIGEYOSHI OWA Using (2.3), we obtain that 2 P ( 2γ)w(z) P α = + w(z) zw (z) ( ( 2γ)w(z) w(z) ( 2γ)w(z) + w(z) ). (3.7) + w(z) If we suppose that there exists a point z 0 U such that then Lemma 2 gives us that max w(z) = w(z 0) = (w(z 0 ) ), z z 0 z 0 w (z 0 ) = kw(z 0 ) (k ). Therefore, letting w(z 0 ) = e iθ (θ π), we have P f(z 0 ) ( 2γ)w(z0 ) P α = f(z 0 ) + w(z 0 ) k ( ( 2γ)w(z0 ) 2 ( 2γ)w(z 0 ) + w(z ) 0) = + w(z 0 ) ( 2γ)e iθ = + e iθ k ( ( 2γ)e iθ ) eiθ + 2 ( 2γ)eiθ + e iθ = = γ k ( ( 2γ)(cos θ ( 2γ)) 2 + ( 2γ) 2 2( 2γ) cos θ + ) 2 γ k ( 2 2 2γ ) γ(3 4γ) =. (3.8) 2( γ) 4( γ) This contradicts our condition (3.2). Therefore, we have P α w(z) = P α P α < (z U), (3.9) P α + ( 2γ) which implies (3.3). Taking = 0 and = /4 in Theorem 3, we have Corollary 4. Let be in the class A. Then P P α > 0 (α > 2; z U) P α = P α > 3 4 (z U)

9 and PROPERTIES OF CERTAIN INTEGRAL OPERATORS 543 Finally, we prove P P α > (α > 2; z U) 4 P α = P α > 2 Theorem 4. If A satisfies for some γ (γ < ), then (z U). > γ (α > 2; > ; z U) (3.0) α Q α > δ (z U), (3.) where δ (0 δ < ) is the smallest positive root of the equation 2(α + )δ 2 2(α + )(γ + ) (2γ )δ + 2(α + )γ + = 0. (3.2) Proof. Defining the function w(z) by we obtain that Q α = Q = zw (z) w(z) ( 2δ)w(z), (3.3) + w(z) ( 2δ)w(z) (α + ) + w(z) α + ( ( 2δ)w(z) ( 2δ)w(z) + w(z) + w(z) Therefore, supposing that there exists a point z 0 U such that we see that max w(z) = w(z 0) = (w(z 0 ) ), z z 0 f(z 0 ) f(z 0 ) (α + )δ α + Making γ = 0 in Theorem 4, we have δ 2( δ) ). (3.4) = γ. (3.5)

10 544 SHIGEYOSHI OWA Corollary 5. If A satisfies then > 0 (α > 2; > ; z U), (3.6) α Q α > α + (z U). (3.7) Letting γ = /2 in Theorem 4, we have Corollary 6. If A satisfies then > 2 α Q α > α + 3 α + (α > 2; > ; z U), (3.8) (z U). (3.9) Further, if α = 3, we have Q 2 > 2 = ( > ; z U) 2 Q 3 Acknowledgement > 0 (z U). This research was supported in part by the Japanese Ministry of Education, Science and Culture under a Grant-in-Aid for General Scientific search (No ). ferences. S. D. Bernardi, Convex and starlike univalent functions. Trans. Amer. Math. Soc. 35 (969), S. D. Bernardi, The radius of univalence of certain analytic functions. Proc. Amer. Math. Soc. 24 (970), I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 3 (97),

11 PROPERTIES OF CERTAIN INTEGRAL OPERATORS I. B. Jung, Y. C. Kim, and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, to appear. 5. S. S. Miller and P. T. Mocanu, Second-order differential inequalities in the complex plane. J. Math. Anal. Appl. 65 (978), S. Owa and H. M. Srivastava, Some applications of the generalized Libera operator. Proc. Japan Acad. 62 (986), H. M. Srivastava and S. Owa, New characterization of certain starlike and convex generalized hypergeometric functions. J. Natl. Acad. Math. India 3 (985), H. M. Srivastava and S. Owa, A certain one-parameter additive family of operators defined on analytic functions. J. Anal. Appl. 8 (986), Author s address: Department of Mathematics Kinki University Higashi-Osaka, Osaka 577 Japan (ceived )