Sentence examples for modulus f and from inspiring English sources

For any modulus f and positive α, (w_{alpha,o}^{f} subset w_{alpha,infty}^{f}).

For any modulus f and (alphageq1), (w_{alpha}^{f} subset w_{alpha,infty}^{f}).

(i) For any modulus f and positive α, (w_{alpha,o}^{f} subset w_{alpha,infty}^{f}).   (ii) For any modulus f and (alphageq1), (w_{alpha}^{f} subset w_{alpha,infty}^{f}).  .

As expected, finite sets have zero (f_{alpha} -density f_{alpha} -densitymodulus fornd (anyhain( 0,1 ]).

Recently Savaş [14] generalized the concept of strong almost convergence by using a modulus f and examined some properties of the corresponding new sequence spaces.

For any modulus f and positive real number α, the spaces (w_{alpha,infty}^{f}) and (M w_{alpha,infty}^{f})) are normal as well as monotone.

For any lacunary sequence θ, and unbounded modulus f for which (lim_{t toinfty}frac{f(t)}{t} > 0) and there is a positive constant c such that (f xy geq cf(x f y)) for all (xgeq0), (y geq0), we have (mathit{WS}^{f} subset mathit{WS}_{theta}^{f}) if and only if (liminf_{r} q_{r} >1).

For any lacunary sequence θ, and unbounded modulus f for which (lim_{t toinfty}frac{f(t)}{t} > 0) and there is a positive constant c such that (f xy geq cf(x f y)) for all (xgeq0), (y geq0), we have (mathit{WS}_{theta}^{f} = mathit{WS}^{f}) if and only if (1 <liminf_{r} q_{r} leqlimsup_{r} q_{r} < infty).

For any lacunary sequence θ, and unbounded modulus f for which (lim_{t toinfty}frac{f(t)}{t} > 0) and there is a positive constant c such that (f xy geq cf(x f y)) for all (xgeq0), (y geq0), we have (mathit{WS}_{theta}^{f} subset mathit{WS}^{f}) if and only if (limsup_{r} q_{r} < infty).

Many authors, including Connor[20], Kolk[21], Maddox[22], Öztürk et al.[23], Pehlivan et al.[24, 25] and many others used a modulus f to construct some sequence spaces.

For any modulus f such that (lim_{trightarrow infty }frac{f ( t ) }{t}>0) and α̃ ⪰1.