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In this paper, we solve the following quadratic ρ-functional inequalities ‖f (x + y + z/2) + f(x - y - z/2) + f(y - x - z/2) + f(z - x - y/2) -f(x) - f(y) - f(z)‖ ≤ ‖ρ(f(x + y + z) + f(x - y - z) + f(y - x - z) + f(z - x - y) (0.1) -4f(x) - 4f(y) - 4f(z))‖; where ρ is a fixed complex number with |ρ| < 1/8, and ‖f(x + y + z) + f(x - y - z) + f(y - x - z) + f(z - x - y) -4f(x) - 4f(y) - 4f(z)‖ (0.2) ≤ ‖ ρ(f(x + y + z/2) + f(x - y - z/2) + f(y - x - z/2) + f(z - x - y/2) -f(x) - f(y) - f(z))‖ ; where ρ is a fixed complex number with |ρ| < 4. Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following quadratic ρ-functional inequalities ..., where ρ is a fixed complex number with |ρ| < 1/8, and ..., where ρ is a fixed complex number with |ρ| < 4. Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the additive ρ-functional inequalities ‖f(2x - y) + f(y - x) - f(x)‖ ⩽ ‖ρ (f(x + y) - f(x) - f(y))‖ ; (0.1) where ρ is a fixed complex number with ¦ρ¦ < 1, and ‖f(x + y) - f(x) - f(y)‖ ⩽ ‖ρ(f(2x - y) + f(y - x) - f(x))‖k; (0.2) where ρ is a fixed complex number with ¦ρ¦ < 1/2. Furthermore, we prove the Hyers-Ulam stability of the additive ρ-functional in-equalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the quadratic ρ-functional equations f (x+ y)+ f (x− y)− 2f (x)− 2f (y)(0.1)= ρ(2f(x+y/2)+ 2f(x−y/2)− f (x)− f (y)),where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ≠− 1, 2,and2f(x+y/2)+ 2f(x−y/2)− f (x)− f (y)(0.2)= ρ (f (x+ y)+ f (x− y)− 2f (x)− 2f (y)),whereρ is a fixed non-Archimedean number or a fixed real or complex number with ρ≠− 1, 1/2.Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functionalequations (0.1) and (0.2) in non-Archimedean Banach spaces and in Banach spaces. [ABSTRACT FROM AUTHOR]

Let M1f(x; y) : = 3/4f(x+y) - 1/4 f(-x - y) + 1/4 f(x - y) + 1/4 f(y - x) - f(x) - f(y); M2f(x; y) := 2f (x + y/2 )+ f (x - y/ 2 )+ f (y - x/ 2 )- f(x) - f(y). We solve the additive-quadratic ρ-functional inequalities ‖M1f(x; y)‖ ≤ kρM2f(x; y)‖, (0.1) where ρ is a fixed complex number with ∣ρ∣ < 1/2 and ‖M2f(x, y)‖ ≤ kρM1f(x, y)‖, (0.2) where ρ is a fixed complex number with ∣ρ∣ < 1. Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the quadratic ρ-functional equations f(x + y) + f(x - y) - 2f(x) - 2f(y) (0.1) = ρ ( 2f ( x + y/2 ) + 2f ( x - y/2 ) - f(x) - f(y)), where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ≠ -1, 2, and 2f ( x + y/2 ) + 2f ( x - y/2 ) - f(x) - f(y) (0.2) = ρ(f(x + y) + f(x - y) - 2f(x) - 2f(y)), whereρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ≠ -1; 1/2 . Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional equations (0.1) and (0.2) in non-Archimedean Banach spaces and in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following additive-quadratic ρ-functional inequalities ‖f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y)‖ (0.1) ⩽ ǁρ 2f ( x + y 2 ) + 2f ( x - y 2 ) - 3/2 f(x) + 1 2 f(-x) - 1/2 f(y) - 1/2 f(-y)ǁ ; where ρ is a fixed complex number with ǀρǀ < 1, and ǁ2f ( x + y 2 ) + 2f (-x - y 2) - 3/2 f(x) + 1 2 f(-x) - 12 f(y) - 1/2 f(-y)ǁ ⩽‖ρ(f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y)ǁk; (0.2) where ρ is a fixed complex number with ǀρǀ < 1 2 , and prove the Hyers-Ulam stability of the additivequadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive-quadratic ρ-functional equations associated with the additive-quadratic β-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following cubic ρ-functional inequality ‖f(2x + y) + f(2x - y) - 2f(x + y) - 2f(x - y) - 12f(x)‖ ⩽ ǁ ρ 4f (x + y/2 )+ 4f (x - y/2) - f(x + y) - f(x - y) - 6f(x))ǁ ; (0.1) where ρ is a ρ xed complex number with ǀρǀ < 2, and the quartic ρ-functional inequality ‖f(2x + y) + f(2x - y) - 4f(x + y) - 4f(x - y) - 24f(x) + 6f(y)‖ (0.2) ⩽ ǁ ρ(8f x + y/2( + 8f (x - y/2) - 2f(x + y) - 2f(x - y) - 12f(x) + 3f(y) ǁ ; where ρ is a fixed complex number with ǀρǀ <2. Using the direct method, we prove the Hyers-Ulam stability of the cubic ρ-functional in- equality (0.1) and the quartic ρ-functional inequality (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following additive-quadratic ρ-functional inequalities ... where ρ is a fixed complex number with |ρ| < 1, and ... where p is a fixed complex number with |p| < 1/2, and prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive-quadratic p-functional equations associated with the additive-quadratic p-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following cubic p-functional inequality Il f (2x + y) + f (2x - y) - 2f (x + y) - 2f (x - y) - 12f (x)|| where p is a fixed complex number with | ρ| < 2, and the quartic ρ-functional inequality ||f (2x + y) + f (2x - y) - 4 f (x + y) - 4f (x - y) - 24f (x) + 6f(y)||... where ρ is a fixed complex number with | ρ| < 2. Using the direct method, we prove the Hyers-Ulam stability of the cubic ρ-functional inequality (0.1) and the quartic ρ-functional inequality (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the following new additive s-functional inequalities ||f(x - y) + f(y) + f(-x)|| ≤ ||s (f(x + y) - f(x) - f(y)) k; (0.1) ||kf(x + y) - f(x) - f(y)|| ≤ ||s (f(x - y) + f(y) + f(-x)) k; (0.2) where s is a fixed complex number with |s| < 1, and prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequalities (0.1) and (0.2). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the following new additive s-functional inequalities ‖f(x - y) + f(y) + f(-x)‖ ≤ ‖s (f(x + y) - f(x) - f(y)) ‖; (0.1) ‖f(x + y) - f(x) - f(y)‖ ≤ ‖s (f(x - y) + f(y) + f(-x)) ‖; (0.2) where s is a fixed complex number with |s| < 1, and prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequalities (0.1) and (0.2). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the following new additive s-functional inequalities ‖f(x - y) + f(y) + f(-x)‖ ≤ ‖s (f(x + y) - f(x) - f(y)) ‖; (0.1) ‖f(x + y) - f(x) - f(y)‖ ≤ ‖s (f(x - y) + f(y) + f(-x)) ‖; (0.2) where s is a fixed complex number with |s| < 1, and prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequalities (0.1) and (0.2). [ABSTRACT FROM AUTHOR]

Let n, k be positive integers satisfying n ≥ 2k + 4, a ≠ 0 be a complex number, and let F be a family of functions meromorphic in a domain D, if for each f ∊ F, fn + af(k) and gn + ag(k) share b, and all zeros have multiplicity at least k + 1, then F is normal in D. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the additive (α β)-functional equation f(x) + αf(αy) = β-1f(β(x + y)) for all complex numbers α with ¦α¦ = 1 and for a fixed nonzero complex number β. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (0.1) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we de ne and study a function F : [0,∞) → R and extensions F: R→C, F : C → C which are continuous and such that if n ϵ Z, the set of all integers, then F(n)=Fn, the nth Fibonacci number based on F0 = F1 = 1. If x is not an integer and x < 0, then F(x) may be a complex number, e.g., F(-1.5) = 1/2+i. If z = a + bi, then e F(z) = F(a) + iF (b - 1) defines complex Fibonacci numbers. In connection with this function (and in general) we define a Fibonacci derivative of f : R → R as (▵f)(x) = f(x + 2) - f(x + 1) - f(x) so that if (▵f)(x) ≡ 0 for all x ϵ R, then f is a (real) Fibonacci function. A complex Fibonacci derivative ▵ is given ▵ f(a + bi) = ▵ f(a) + i ▵ f(b - 1) and its properties are discussed in same detail. [ABSTRACT FROM AUTHOR]

Let M1f(x; y) : 3/4f(x+y)-1/4f(-x-y) + 1/4f(x-y)+1/4f(y-x)+1/4f(y-x)-f(x)-f(y), M2f(x,y): = 2f(x+y/2)+f(x-y/2)+f(y-x/2)-f(x)-f(x). We solve the additive-quadratic ρ-functional inequalities ‖M1f(x,y)‖⩽‖ρM2f(x,y)ρ, where ρ is a fixed complex number with |ρ| < 1 Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

Let M1f(x; y) : 3/4f(x+y)-1/4f(-x-y) + 1/4f(x-y)+1/4f(y-x)+1/4f(y-x)-f(x)-f(y), M2f(x,y): = 2f(x+y/2)+f(x-y/2)+f(y-x/2)-f(x)-f(x). We solve the additive-quadratic ρ-functional inequalities ‖M1f(x,y)‖⩽‖ρM2f(x,y)ρ, where ρ is a fixed complex number with |ρ| < 1 Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

M1f(x,y): = 3/4f(x+y)-1/4f(-x-y) +1/4f(x-y)+1/4f(y-x)-f(x)-f(y), M2f(x,y):=2f(x+y/2)+f(x-y/2)+f(y-x/2)-f(x)-f(y). We solve the additive-quadratic ρ-functional inequalities ‖M1f(x,y)‖≤‖ρM2f(x,y)‖≤‖ρM1f(x,y)‖, where ρ is a fixed complex number with ∣ρ∣<1. Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the additive ρ-functional equations ... where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ≠ 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ- functional equations (0.1) and (0.2) in non-Archimedean Banach spaces and in Ba- nach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the additive ρ-functional equations f(x + y) - f(x) - f(y) = ρ(2f(x + y/2)-f(x)-f(y)), (0.1) 2f(x + y) - f(x) - f(y) = ρ (f(x + y) - f(x) - f(y)) ; (0.2) where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ≠1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ- functional equations (0.1) and (0.2) in non-Archimedean Banach spaces and in Ba- nach spaces. [ABSTRACT FROM AUTHOR]